By Karey Wutkowski

WASHINGTON, May 22 (Reuters) - The U.S. Treasury Department could offer capital matching for small banks that would help remove any stigma from receiving government aid, the chairman of the Federal Deposit Insurance Corp said on Friday.

Sheila Bair said she has proposed that Treasury create a program within the $700 billion Troubled Asset Relief Program (TARP), in which the government would match any private capital that small banks raise.

Bair said she is concerned that a lot of capital is flowing to the 19 largest U.S. banks that were recently stress tested by the government and that community banks are being left out.

"I do worry that we are putting smaller banks at an increasingly vulnerable position," Bair told reporters before an FDIC board meeting.

Bair said the banks that really need TARP funds are not going to ask for them, noting the stigma associated with accepting government money.

She said a dollar-for-dollar matching program could be a better option as Treasury reopens its capital injection program with a focus on smaller banks.

"We've suggested it, and I think, I hope they're looking at ways we can make the program work better for smaller banks," Bair said.

A Treasury spokesman did not immediately respond to a request for comment.