The Gunners rejected a loan offer from the Saints earlier this month, while Wolves have also shown interest in the England goalkeeper.
But Southampton have since returned to the negotiating table with Arsenal, who may be willing to reduce the purchase price at the end of the loan period.
Southampton initially wanted to strike a deal that included an option to buy Ramsdale next summer.
However, Arsenal have been pushing for the inclusion of an obligation for the Saints to make the deal permanent.
A possible compromise could see the two clubs agree that Southampton would be obliged to sign Ramsdale for a set price next summer only if they stay up this season.
Ramsdale became Arsenal's most expensive goalkeeper ever when he arrived from Sheffield United for a transfer fee of £30million in August 2021.
The 26-year-old has since made 89 first-team appearances for the north London club.
He only featured in six Premier League matches last season though, after losing his status as Arsenal's No 1 to David Raya.
Ramsdale has been capped five times by England at senior international level.
He was part of the Three Lions squad that went to Euro 2024 but he did not get on the pitch in Germany.
England international Ramsdale has been a regular on the Arsenal bench so far this season, having featured in just six league games in the previous campaign
Irish stopper Gavin Bazunu was Southampton's first-choice keeper in the Championship last season but he ruptured his Achilles tendon in April and will not be fit again until 2025.
In his absence, Alex McCarthy has started the first two games of the current campaign.