Mourinho is keen to make at least two more additions to his Manchester United squad before the transfer window closes on August 9.
But the Portuguese boss recently admitted there has been little movement in their negotiations for players.
After an impressive World Cup for England, Maguire has been linked with a move to Old Trafford but Leicester’s asking price would be around £65million.
According to the Manchester Evening News, Mourinho has been interested in Alderweireld and Boateng but he has urged executive vice-chairman Ed Woodward to sign Maguire instead.
It is claimed Mourinho is hopeful the prospect of Champions League football is enough to convince Maguire to snub a new contract from Leicester and join United.
Maguire joined Leicester a year ago from Hull City and has gone from strength to strength.
But Foxes boss Claude Puel is desperate to keep hold of the centre-back and took the positives from the interest in him.
He said: “It is important for Maguire to continue for Leicester.
“It is a good feeling. If our players are interesting a lot of clubs it is because we are recruiting well and doing good work. It is important we continue this.
“When we welcome players back from the World Cup, I hope we can give them the opportunity to continue advancing their ambitions and perform in the Premier League.”