Wissa has made clear he wants to leave for St James’ Park and has not been training or playing with the first team this season.
According to sources, Brentford originally wanted £40m for Wissa but the price has now increased to about £65m.
The transfer window closes at 19:00 BST on Monday.
Brentford have already lost fellow forward Bryan Mbeumo this summer to Manchester United for £65m, but signed Dango Ouattara from Bournemouth for a club record deal worth up to £42.5m.
Wissa’s possible move to Newcastle is believed to be part of a series of transfers that could result in Alexander Isak joining Liverpool.
Isak is in a similar situation at Newcastle, accusing the club of “broken promises” as he tries to force an exit to the Premier League champions.
In his statement, Wissa – who has scored 49 goals in 149 games since he joined in 2021 – said: “I have stayed silent for much of the summer, but with just hours remaining of the transfer window I feel compelled make it clear that I want to leave Brentford. I believe the club are unduly standing in my way despite a series of fair offers throughout the summer.
“This has left me in a difficult and frustrating position. It saddens me to have to write that, and I maintain total respect for the club and its fans.”
He added: “I ask Brentford’s owners and directors to now honour their promise to let me leave in the final hours of the window.”
BBC Sport reported in July that Wissa flew home early from the club’s summer training camp in Portugal after making clear he wanted the move to Newcastle.
Ever since he has been in a stand-off with the club over his intention to join Eddie Howe’s side.
He did return to first-team training for a short period but has gone back to working away from Keith Andrews’ squad.
Wissa, who scored 19 goals in the Premier League last season, also removed all association with Brentford from his Instagram account.
He says a formal offer was submitted by another Premier League club and he was “under the impression, from all my conversations with Brentford, that there was a mutual agreement to part ways”.
He added: “I want to make it clear that I have not acted unprofessionally, nor do I wish to leave Brentford on bad terms. I have been transparent in my position throughout. I have continued to communicate openly with the club and conduct myself in a way that reflects my values as both a footballer and a human being.
“In the meantime, I must do what I feel is right for my career and family and insist that Brentford honour their promise to let me join a new club and at a fair price.”
BBC Sport has contacted Brentford for comment.