Afghan officials approved an agreement in November on terms for the continued presence of U.S. troops but Karzai has said he will not sign the agreement until after the April 5 presidential election.
The White House had planned for President Barack Obama to announce a decision during Tuesday's State of the Union address on pulling out of Afghanistan, the Hill reported.
Vice President Joe Biden and other administration officials have lobbied for leaving a residual force of 2,000- to 3,000 troops in Afghanistan after the major withdrawal while U.S. military officials have advocated leaving 10,000 troops in the country.
U.S. Sen. Claire McCaskill, D-Mo., told the Hill Karzai is "divorced from reality at this point," and Sen. Kelly Ayotte, R-N.H., said Karzai's public suggestion that U.S. troops are responsible for insurgent attacks in Afghanistan "makes me deeply concerned that he's almost becoming delusional in terms of what the U.S. role is there."
"The best thing to do is to wait until after the elections with a new president," said Sen. John McCain, R-Ariz., a member of the Senate Armed Services Committee.
Committee Chairman Carl Levin, D-Mich., said Washington "shouldn't just be acting as though Karzai's essential. He's not essential."