Scott felt discomfort before Saturday's 7-0 Nations League win over Croatia but started alongside Elliot Anderson and played 70 minutes before being substituted. He has since withdrawn from England's squad to face the Czech Republic on Tuesday and returned to Bournemouth for treatment.
Thomas Tuchel defended England's handling of the 23-year-old, insisting there had been no unnecessary risk after the midfielder had trained fully before the Croatia game.
"We knew he was in treatment but there were only green lights from him and the medics," Tuchel said. "When we were aware there was slight discomfort during the match and asked if he wanted to come off he said no."
Tuchel added that Scott told staff after the match that he could have continued for the full 90 minutes and did not feel injured. However, the subsequent assessment revealed a quadriceps problem that could keep him out for several weeks.
Bournemouth are understood to be frustrated because England knew Scott had been carrying the issue before he played against Croatia. The potential six-to-eight-week absence would be a major blow for Marco Rose's side, who are scheduled to play nine matches between 10 October and 8 November.
Those fixtures include Premier League games against Chelsea, Sunderland, Manchester United, Leeds and Ipswich, as well as Europa League meetings with Sturm Graz, AC Milan and Sparta Prague and a Carabao Cup tie against Aston Villa.
Scott had only just established himself in Tuchel's England plans. He made his senior debut in the 2-0 win over the Czech Republic before earning his first start against Croatia, where his performance drew lavish praise from the England manager.
"First start and what a start, what a performance," Tuchel said. "He is so smooth on the ball, his decision making is at the highest level."
Tuchel also described Scott as "the complete package" and said his emergence gave England greater variety in midfield.
The injury therefore comes at a particularly frustrating time for Scott, who had begun to strengthen his case for a regular international role. For Bournemouth, however, the immediate concern is his availability during a demanding run of domestic and European fixtures.
Scott joins Eli Junior Kroupi on Bournemouth's injury list, while the club continue to seek consistency after a difficult start to the Premier League season.