It has to happen before the congressmen go on recess.
US Undersecretary of State Wendy Sherman told a meeting of the International Student House in Washington on Tuesday, July 12, that the White House expects congressmen to approve a protocol for Finland and Sweden to join NATO before they leave for recess in August.
"I hope that the US Congress will ratify [Finland and Sweden's accession to NATO] before they go on recess in August. Given everything else that's on their agenda, I'm not sure they'll succeed, but I'm hopeful," Sherman said.
She also suggests that the rest of the alliance countries "are moving very quickly on this as well."
Note that the upper chamber of the US Congress will go into recess on August 8 for almost a month. It is the Senate which has to give its consent to NATO enlargement. President Joe Biden sent the protocol on admitting Finland and Sweden to the alliance to the upper house for consideration on July 11.