Welker, who underwent reconstructive left knee surgery over the offseason, has been participating in full-contact drills during the team's training camp. He fully participated in joint practices with the Saints on Tuesday and Wednesday.
"We'll see how it goes here yesterday and then today and make a decision on that," said Belichick. "I don't know. I'm not saying he will; I'm not saying he won't. We'll evaluate it and see."
Welker had told ESPN NFL insider Adam Schefter on Friday that he hopes to be ready to play in the first game of the season against the Bengals on Sept. 12.
"My goal is to get out there on the field as soon as possible," Welker said, "and Week 1 is definitely something I'm shooting for. We'll see what happens."