HITC Sport understands that the pair, a midfielder and a winger, have linked up with the Royals on a trial basis.
The Championship outfit have been following both players closely in recent months and have now moved to fend off interest from a number of other Football League clubs.
Both Dulwich Hamlet midfielder Sweeney and Tower Hamlets winger Williams met with Reading earlier this month, and HITC Sport understands they have both now linked up with the club for training.
Both 21-year-old Sweeney and Williams, 18, are on non-contract terms with their current clubs so Reading are free to talk about bringing them in on a trial basis.
Sweeney, who only joined Dulwich a matter of weeks ago from Kingstonian, has been in superb form, scoring seven times in as many games, and he had more than a dozen clubs chasing him – but now Reading are in pole position.
The Royals are also poised to land Williams, a former Tottenham Hotspur youth trainee who had drifted into non-league – but he has delivered some stunning displays which have impressed scouts and now he has the chance to seal a switch to the Madjeski Stadium.