The 28-year-old left-back has only managed 15 appearances in an injury-blighted season, with his last game coming three months ago in a win at Luton.
Shaw had been expected to return from his latest muscle problem in early May, but he is now unlikely to feature in the FA Cup final against Manchester City on May 25.
With England boss Gareth Southgate set to name his provisional Euro 2024 squad just a few days after the showpiece match at Wembley, Shaw is facing a race against time to prove his fitness for the tournament.
Shaw's return from his latest muscle injury was due in early May, but Ten Hag recently revealed the Euro 2020 final goalscorer had suffered a setback in his rehabilitation.
On Thursday, Ten Hag was asked about the Euro 2024 and FA Cup final fitness hopes of Shaw and Harry Maguire, who has missed three matches with a muscle complaint of his own.
"For England I can't say (if they are available), it's not up to me," Ten Hag told reporters.
"But for the cup final, we're working on (it) and I think Harry Maguire has a fair chance that he will be available. We plan (for) him.
"Luke is more complicated. In this moment, let's say it's a less chance that he will make it but there is still a very small, reduced chance.
"He had the setback and now we have to find out how far he is."
England play warm-up friendlies against Bosnia-Herzegovina on June 3 and Iceland on June 7, before travelling to Germany, where they begin their Euro 2024 campaign against Serbia on June 16.
