Robert Smith's band are rumoured to be in line to play the double-header over the August Bank Holiday Weekend, Holy Moly reports.
The website claimed that the festival shows will be The Cure's first-announced UK dates of 2012.
Reading and Leeds organisers confirmed last month that they would unveil the headline acts and a majority of the supporting bill on March 12 at 7.15pm.
The Cure last year played a career-spanning set at Bestival, which was later released as a live album for charity.
The band recently confirmed festival shows in Sweden and Germany.
Watch The Cure perform hit single 'Close To Me' live at Bestival below: