The 2016 PFA Player of the Year failed to find a route out of the King Power Stadium this summer.
Having made his intentions clear at the end of last season he wished to go, only Roma came in with offers.
Leicester rejected their approaches which were way below their valuation of Mahrez.
Chelsea, Arsenal and Tottenham are all admirers but did not pursue a deal.
Mahrez was even granted permission by Algeria to leave on international duty and secure a move, but he eventually stayed on deadline day.
According to the Daily Mail, the 26-year-old will now target a January departure.
Leicester ultimately priced him out of a move by demanding £50m.
Roma went as high as £32m but it was not enough as the Foxes held firm, leaving Mahrez frustrated.
However, he has continued to remain professional on the pitch and has shone so far this season.
But the Daily Mail add Mahrez is aware Chelsea, Arsenal and Spurs will only do business in January if he shows the same form from when Leicester won the Premier League.
The playmaker had less of an impact last term but has caught the eye in his early displays.
Reports on deadline day suggested Chelsea were in for Mahrez, but a move failed to materialise as they signed his Leicester team-mate Danny Drinkwater instead.
Arsenal failed to strengthen beyond Alexandre Lacazette and Sead Kolasinac.
Meanwhile, Spurs were busy at the end of the window but stayed away from Mahrez.
Barcelona would be a dream move for Mahrez, but a switch to one of the Premier League's elite would also appeal.
