The Gunners reportedly saw a £70M bid rejected by the Magpies yesterday, after weeks of speculation over their intentions on the 28-year-old, following his rumoured indication of wanting to leave.
Arsenal 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.
That first offer is believed to be in the region of £70M up front - followed by £10M in performance related instalments - but that does not match Newcastle's current valuation of their skipper.
Guimarães remains with the squad in their preseason training camp in Spain and Newcastle are staying strong over their position of wanting to keep him at St. James' Park - with two years left on his contract and a club-held option for a third.
As per Telegraph Sport, Arsenal are now willing to offer £80M as a stand alone fee, but that is the celling for Mikel Arteta's defending champions as they await a final response from Newcastle.
Newcastle take on Valencia in friendly action on August 8th and Guimarães could feature if an agreement has not ben reached.