Rodriguez, who was the leading earner at Goodison Park, left the Toffees for Qatari side Al-Rayyan on Wednesday, freeing up a large section of the wage bill in Merseyside.
According to Daily Mail, Rafael Benitez's side are now considering a move for right-back Serge Aurier, following the Ivorian's contract termination with Tottenham Hotspur, with a section of the funds relieved after Rodriguez's exit.
Such news follows the hamstring injury to captain Seamus Coleman, who had featured in the first four games of the season on the right-hand side of the defence before being sidelined while away on international duty.
With no clear deputy in the first-team squad, centre-back Ben Godfrey has filled in on the right-hand side in Coleman's absence.
Aurier made 110 appearances for Tottenham over a four-year period, before departing prematurely ahead of the new campaign.