FThe Gunners have been tracking the Brazil international since the start of the summer and they are looking to seize on uncertainty at St. James' Park following Eddie Howe's departure as Newcastle boss.
Howe had previously admitted there has been transfer interest in Guimarães with Arsenal readying a bid for the 28-year-old who has told Newcastle he wants to leave.
As per the latest from Sky Sports, Newcastle have received an offer from the Emirates Stadium - believed to be in the region of £70M up front - followed by £10M in performance related instalments.
That has been turned down by the club's ownership and Guimarães remains with the squad in their preseason training camp in Spain.
Arsenal are now expected to go away and consider a second offer with Newcastle firm on their position of wanting their captain to stay - as he has two years left on his contract- plus a club-held option for a third.
If Newcastle relent over an exit, the club could push for a fee similar to the £100M Tottenham paid them for Sandro Tonali earlier this summer.