Verbal proposals of £55million, and later £65million including add-ons, were both rejected by Newcastle United as talks between the clubs have gathered pace.
Personal terms are not considered an obstacle - Guimaraes, 28, has made clear to Newcastle that his preference is to join the Premier League champions this summer - but sources close to St James' Park continue to insist there has been no direct approach and no intention to sell.
Arsenal are working to a hardline maximum valuation of their own, however, and are not prepared to move significantly beyond the £70million mark regardless of how badly they want the deal done.
That discipline was already on display earlier this window, when the club stuck to an £80million valuation of Morgan Rogers rather than match Chelsea's eventual £117million offer for the Aston Villa forward, even after being given the chance to do so.
Guimaraes would represent a marquee addition to Mikel Arteta's midfield as Arsenal continue reinforcing a squad they hope can retain the title this season. He is not the only Newcastle-adjacent name on the club's radar this summer, either - Arsenal have also identified Aston Villa's Ezri Konsa as their first-choice defensive target, and have registered interest in the availability of Como centre-back Jacobo Ramon, though contractual clauses are understood to complicate any move for the Spaniard.
Guimaraes has been a central figure at St James' Park since arriving from Lyon, helping Newcastle return to the Champions League and establishing himself as club captain, a status that only adds to the difficulty of any potential sale.
Arteta is understood to view the Brazilian's combination of ball retention, tempo control and defensive work as an ideal fit for the way Arsenal want to play. Whether Newcastle can be persuaded to part with such a central figure, even for a substantial fee, remains the key question hanging over the rest of the window.