Manchester United boss Mourinho signed Grant earelier this week with the goalkeeper signing on a free transfer from Stoke.
And United have been on the lookout for a suitable third-choice to complement David de Gea and Sergio Romero after deciding to give Pereira the opportunity to play games elsewhere.
The 35-year old started off his career at Derby County in 2000 and has had subsequent spells with Burnley, Oldham Athletic (on loan), Sheffield Wednesday.
"I hope I can stay in the Premier League".
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The main motive for Manchester United signing the 35-year old is understood to be the fact that Mourinho wants promising young goalkeeper, Joel Pereira, to spend the next season out on loan and continue his development.
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Shaqiri has emerged as one of Jurgen Klopp's potential signings this summer following Stoke City's relegation from the Premier League last season.
Argentina worldwide Sergio Romero is De Gea's understudy, while Joel Perreira is now third-choice under the Portuguese head coach.
Lee Grant will undergo a medical at Manchester United at Old Trafford later this week, according to Dharmesh Sheth.
The toxic and unfriendly atmosphere in Spain will surely do little to tempt De Gea toward Madrid, and with a new five-year-deal reported to be close by the Mirror, the Old Trafford faithful can expect to see him between the posts moving forward.
Both players were only bit-part players at their respective clubs last season.