The Valencia centre-back was thought to be set to join Arsene Wenger's side at the end of the season, but Guus Hiddink's team are reportedly willing to pay up to £13m for Rami as a replacement for Christopher Samba.
The former-Blackburn defender left Anzhi for Queens Park Rangers in the January transfer window, and the Russian club are believed to be willing to shell out £3m a year in salary alone to pip Arsenal to the signing.
The 27-year-old has been a long-term target for the Gunners, with Wenger thought to be keen to bolster his defensive options after a mixed campaign so far.
Arsenal have kept back-to-back clean sheets in their last two matches against Swansea and Bayern Munich, but over the course of the season they've given up sloppy goals in key moments of big games.
Rami joined Valencia from Lille in the summer of 2011 after winning the Ligue 1 title with the French club.
He has 25 caps for his national team, and appeared for France at Euro 2012. The defender primarily operates at centre-back but has been known to occasionally step up into midfield.