Weeden could be ready to practice and play as early as next week, the source said.
Browns coach Rob Chudzinski said Wednesday that third-string quarterback Brian Hoyer will start Sunday's game against the Minnesota Vikings. Hoyer will be the 19th quarterback to start for the Browns since 1999.
Weeden injured his thumb when he banged it on the helmet of teammate John Greco during the fourth quarter of Sunday's 14-6 loss to Baltimore.
"You never want to get hurt," said Weeden, who was sacked 11 times in Cleveland's two losses. "I don't care when it is, could be the 16th game or the first game, as a player, you want to be out there. So yeah, it is frustrating. Even though the results weren't there offensively, we were starting to do some better things."
Weeden finished 21-of-33 for 227 yards against the Ravens, but Cleveland's offense sputtered for the second week in a row. The Browns missed chances to make big plays, none bigger than when Weeden missed running back Chris Ogbonnaya wide open down the left sideline.
ESPN NFL Insider Adam Schefter and The Associated Press contributed to this report.