Injuries to both Nacho Monreal and Sead Kolasinac before the season has even begun led to reports of an alleged Gunners bid for Rodriguez earlier this week.
That £30m offer is thought to have been rejected, with reports from Italy now explaining that Rodriguez has no interest in leaving Milan and Serie A for anyone.
It's not simply that he didn't want to join Arsenal because Paris Saint-Germain are also rumoured to have been knocked back in their own efforts to sign the 25-year-old.
Rodriguez, who made the switch from Wolfsburg in Germany last summer, is said to be excited by the 'new project' at Milan being led by new technical director Leonardo and club legend Paolo Maldini, who has taken up a new role as sporting strategy and development director.
With only speculative rumours lacking real substance now doing the rounds for Arsenal, it may be that the Gunners have to be satisfied with their transfer haul this summer and go into the new 2018/19 Premier League season as things are.
Kolasinac is expected to be out for a number of weeks with a leg injury. Monreal, meanwhile, may only miss the opening game against Manchester City with his knee problem.
With no other leftbacks in the first-team Unai Emery will have to think of a more creative solution, for example by relying on the experience of new right-back Stephan Lichtsteiner or the versatility of talented midfielder Ainsley Maitland-Niles to fill in.