The French giants have been chasing the Senegal international since the January transfer window and are increasingly confident that they will this week get their man.
Gueye, who is under contract to Everton until June 2022, is keen to iron out his future after suffering heartbreak in the Africa Cup of Nations final, losing 1-0 to Algeria.
PSG are putting together a deal which will guarantee Everton £30m plus another £10m in bonuses in a bid to push through the transfer.
It will surpass the £35m bid that they had rejected in January and is a clear sign that they aren’t giving up on the signing anytime soon.
Gueye, who turns 30 in September, will be offered a five-year contract to end his three-year stay on Merseyside following his £7.1m move from Aston Villa.
Everton recently signed Fabian Delph in midfield, welcoming the Englishman to Goodison Park from Manchester City for an initial fee of around £8m.
But Marco Silva has dismissed suggestions that Delph is a direct replacement for Gueye, given that he can play in the defensive midfield role.
“I will answer direct to this question,” Silva told the Liverpool Echo when asked if Delph would replace Gueye, who joined Villa as his fellow midfielder moved to Man City.
“They are two completely different players. Gana [Gueye] plays in one role, Fabian can do the role of Gana but he doesn’t come to our squad to replace Gana.
“Gana is still our player and, of course, if we lose Gana then we have to sign a player to replace him because you are talking about one player playing almost all of the games in our squad.
“If it will happen then we have to replace him, it is not because we have signed Fabian Delph. For me he is a box-to-box player and Fabian performs better when he is a box-to-box player.
“Of course, he can play as one of the two midfielders with an offensive midfield player, the role we are giving to Gylfi [Sigurdsson] and one that Bernard and Tom [Davies] can also play there and even Andre [Gomes] because he did in Valencia, and so Fabian can play in that role in our 4-2-3-1 formation we played last season.
“He can also play as an offensive midfielder and can give us more opportunities than we had last season.
“When I said he can bring different solutions it is to play a 4-3-3 with two offensive midfielders, plus one more positional midfielder.
“But the position I can see him performing well is as a box-to-box midfielder player because he has this capacity to do that.”
Aside from Delph, Everton have also signed goalkeeper Jonas Lossl and midfielder Andre Gomes this summer, making the latter’s loan move from Barcelona permanent for around £22m.