Arsenal had two earlier bids knocked back before landing on a figure Newcastle are prepared to accept for their captain, plus bonuses still to be finalised.
The 28-year-old scored nine goals and added five assists in 29 league appearances for Newcastle last season, and has been strongly linked with a move to the Premier League champions for weeks, with personal terms already understood to be agreed between the two parties.
Talks accelerated on Thursday when transfer journalist Ben Jacobs reported that Arsenal were in advanced discussions, with the club said to be optimistic about closing a deal worth more than £75m. L'Equipe's figure of €90m equates to roughly £77m, or around 988 million Swedish kronor.
Only smaller details are said to remain before the transfer is finalised, including the structure of any bonus payments. Guimarães is expected to sign a five-year contract with Arsenal running until the summer of 2031, in what would be a marquee addition to Mikel Arteta's midfield as the club looks to build on last season's title win.
Lyon, who sold the midfielder to Newcastle for around €40m in January 2022, stand to receive more than €8m through a 20% sell-on clause included in that original deal.
The move would mark the departure of the last of Newcastle's title-winning midfield trio, following the sales of Sandro Tonali to Tottenham and Anthony Gordon to Barcelona earlier this window, deals worth close to £100m and £70m respectively.
Newcastle United have already moved to address the gap left behind, bringing in Sean Steur from Ajax and Aladji Bamba from Monaco, though replacing their captain's leadership and the 195 appearances he has made since arriving at St James' Park will be a considerably harder task.