The Swans had initially appeared to be in pole position to land the 19-year-old after having an offer of around £100,000 accepted, but they were last week told to up that bid following interest from Merseyside giants Everton and Liverpool, and Rodgers had appeared downbeat on his hopes of bringing Donnelly to the Liberty Stadium.
But the south Wales club have had another offer accepted by the Belfast outfit, and Donnelly, who it is understood always favoured a move to Swansea, is set to complete his transfer.
Rodgers said: "I think we now have an agreement on both sides in terms of Cliftonville as a club and personal terms with the player, so we hope to complete that fairly soon.
"He will be over here for the game against Tottenham on Saturday. We will look to complete the medical (on Friday) and get that put to bed and the boy can then come over and work on developing his game.
"I will talk more about him once the deal is done but he is coming here without pressure, he is coming from a part-time level.
"He has shone in the early part of the season and there have been other clubs interested, but we will talk more once he signs the papers."