WASHINGTON—Democrats braced for weeks of delay and uncertainty on their roughly $2 trillion education, healthcare and climate package they had hoped to finish by year end, as efforts faltered to secure the pivotal support of Sen. Joe Manchin (D., W.Va.) for the bill.

President Biden spoke with Mr. Manchin at least twice this week to try bringing him on board with the bill, which will need the support of all 50 members of the Senate Democratic caucus to pass. But Mr. Manchin has so far stood by his central critique of the package: that it temporarily funds programs that Democrats intend to later make permanent, as a way to disguise the full price of its provisions.