What is the polite way to ask a client what he wants?

Such a question is hard to answer, because it depends on various things.
Do you always pass through to the manager, or can you intercept the enquiry and deal with it?

Do you know clients well; are you friendly with some?
Is it a corporate company, which has some specific requirements as to what you can say?

If the client is just very pushy, not giving info to you, would your manager be okay with you saying "Sorry, I have this client who won't give me info; do you want to speak with them?"
At small companies I've worked for, managers are happy to take the call, as clients can be in small numbers.

---

It seems impolite to me to just ask, "What do you want?" Is there a better phrase for this?

Yes, it needs to be worded differently, but your opening line has to somehow be that question - "What do you want?"
Otherwise how do you know how to deal with the call if you do not know what they want? They may or may not want the manager, but you do not yet know.

So use an opening such as:

Hi, how may I help you?

If they give you a name at any point, you can repeat it back to them, but don't ever change it. "Hi, it's Robert here." He's called "Robert", not "Bob" - unless he says he's "Bob", etc.

---

The client sometimes asks to be connected to my manager, but my manager wants me to tell him what a client wants, before I connect the client to him. I am searching for phrases to ask for that.

It sounds like it's okay for the call to be put through to the manager, and you just need details of what the call is about to pass along.

So, if the reply is:

I want to speak with the manager.

You just need to remain polite, and while avoiding committing yourself to anything until you know their exact query, make them feel like they are "going to get what they want." Because they will get want they want, either it turns out you can deal with it and so help them, or the manager is required and they get put through.

So first off, try to "gently" steer them away from wanting to speak with the manager, and not committing yourself to anything, so respond with something like:

Can I ask what it's regarding please? I'll likely be able to help you.

This will allow the client to explain, and using the word "likely" (or probably, etc.) leaves it open to:

Deal with it (if you can) and now the word is "definitely":

Yes, I can definitely help you with that...

Then go on to whatever you need to do, confirm security details, or email the invoice - whatever.

OR

Pass through to the manager if it turns out that is necessary:

Yes, it would actually be the manager, bear with me while...

As they've explained what their call is about, you now know the manager is required and you can pass the details on to your manager.

And the client is now getting exactly what they asked for.

Alternative:
If you always pass through to your manager (never deal with it yourself from such requests) then leave off the "I'll likely be able to help you."

And when you have the info, just say:

Thank you, I'm trying to put you through now.

Keyword here is something non-committing, like "trying." As saying "I'm putting you through now" is committing, and the manager might be unavailable.

---

If they are being pushy and won't explain, just remain calm, polite, and still trying to assure them that they are going to get what they want.

So make it clear that you need info to deal with the call:

Client:

I just want to speak with the manager.

You:

Ok, but I do need to know what your enquiry is so I can pass the details on to the manager.

The "okay" is important, as it's not confirming you're going to put them through, because you then state you need something. But it's a subtle way to make them feel they are now getting what they want, and are likely to explain.

Rather than (e.g.) less positive in their favour like: "Sorry, no, I need info."

If they are still refusing to give info, try not to get frustrated, because the client already is (rightly or wrongly) frustrated and both of you being frustrated will lead to a terrible outcome.

No matter how pushy they are, stand your ground while being polite, and continue to assert that you are not stopping them from speaking with the manager, they just need to explain their details so you can tell the manager.

---

Other info

I think the biggest problem most people have is repeatedly asking the client "please explain," as if they're stopping the client from getting what they want or asked for.
But you are only trying to find out what they really want so you can actually help them appropriately.

Always avoid saying things which are not true (or might not be true), as this generally comes back on you.
Before you know what the call is about, avoid saying things such as, "It would be me who deals with this," as it might turn out your manager or someone else needs to deal with it.

Or pretending the manager is not there. Only do this if you know the manager would not want to speak with them.
What if the manager does need to deal with the call?
Or the client knows the manager is there.

Client was on their mobile, chatting to the manager who was on their office phone, and they got cut off or their battery went flat?
Sure, it's "unlikely," but annoying for a client, especially when the manager will find out, is not worth it.

Just be careful playing around with lies and "trickery" - especially when a friendly and assertive approach can deal with any situation.

Don't ever state where fault lies (e.g. "the manager said I couldn't" or "it's company policy") as this just annoys people.
You always need to make it clear they are going to get what they want, but you need something before that could possibly happen - "I'm happy to help you, but I really need to know..."