The Catalan giants are pursuing a move for Liverpool's playmaker as they seek to replace Neymar.
Ernesto Valverde needs reinforcements in attack to support Lionel Messi and Luis Suarez.
And according to TransferMarketWeb, Leicester offered Barcelona the opportunity to take Mahrez.
The Foxes are holding out for £40m with the 2016 PFA Player of the Year keen to move on.
Roma have submitted one final bid as they look to reach an agreement with Leicester over a deal.
However, the Serie A club are yet to match Leicester's valuation with just over two weeks left to go in the window.
As a result of Barcelona's difficulty in landing either Coutinho from Liverpool or Ousmane Dembele from Borussia Dortmund quickly, Leicester have looked to take advantage.
TransferMarketWeb add, though, Barcelona have turned down the Foxes' proposal as they continue to chase their two top targets.
Barcelona could afford to meet Mahrez's price-tag following Neymar's £200m switch to Paris St-Germain.
But they are determined to land Coutinho and Dembele.
The Brazilian has told Liverpool he wants to leave following a £90m rejected offer from Barcelona.
Reports have also suggested the La Liga club are closing in on a £90m deal for 21-year-old Dembele.
Mahrez, meanwhile, might now stay at the King Power Stadium.
Barcelona are close to spending the first batch of their £198m Neymar windfall.
Midfielder Paulinho is set to have a medical on Thursday after a £36.4m fee was agreed with Guangzhou Evergrande for the ex-Tottenham man.
