Clare, 21, was also the subject of interest from Aston Villa and Leeds United after turning down a new contract at Sheffield Wednesday.
The player met with Swansea's new manager Graham Potter last week for talks about a move.
The deal does not need to be completed by the 5pm transfer window close in England as he is a free agent, so it looks as if the transfer will be finalised on Friday.
Sean Clare spent half of the last campaign on loan with Gillingham in League One but later returned to the 'Owls' and featured five times in the Championship.