Background information:
I am a french citizen already working in NY almost for 2 years with a L1-A visa. Before coming to the US I have been working and living in Europe.
I will be offered a permanent US contract and will be sponsored by my employer to get a green card.
I am planning to buy an apartment in NYC.
In order to buy an apartment I will have to bring my savings in Europe into the US.
My savings in Europe are in the form of bank accounts and stock, which have already paid income tax in Europe.

Question:
If I bring that money (approx. $350.000) into the US, will I have to pay taxes on it? Is there any treaty between the US and Europe to avoid double taxation of capital (income tax in Europe and then income tax again in the US)?

Thank you very much for your help.

There are NO transfer tax in the United States.

You will have to submit Form 3520 at the bank to account for the source of the income, but that is an information return only; no taxes are due.