The band are said to be in final negotiations with organiser Michael Eavis, 75, about performing at the festival next year.
When they recently announced their new tour dates, they were asked if they had left the Glasto dates free to play the festival but they dismissed the idea.
An insider said to UK newspaper Daily Star: 'They haven't put pen to paper yet but it's very close. Take That at Glastonbury would be amazing.'
Band member Robbie Williams, 36, recently rejoined his old group for the first time in 15 years and they will head off on their Progress tour next year.