Mkhitaryan, 27, has been sidelined for more than a month with a thigh injury and sat out both of Armenia's matches during the international break in a bid to return to fitness in time for the Anfield clash.
United had formally written to the Armenian FA to request that Mkhitaryan be excluded from his country's World Cup qualifiers against Romania and Poland so that he could continue his recovery in Manchester, according to the Daily Telegraph.
Left-back Shaw has also been out of action for a month due to a hamstring problem and was not called up by England for the World Cup qualifiers against Malta and Slovenia.
The Sun says both Mkhitaryan and Shaw could be available for United's trip to Liverpool.
Mkhitaryan started his first game for United in the 2-1 defeat against Manchester City on 10 September, but was taken off by Mourinho at half-time after repeatedly losing the ball and struggling to influence proceedings. He aggravated a thigh problem during the encounter and has not played for the club since.
"I didn't change after 20 minutes because I didn't want to destroy the players," Mourinho was quoted as saying by the Guardian after the City match.
"I made a couple of decisions [with the starting line-up] because I thought the individual qualities of certain players would give me what I wanted. But I didn't get it."
Mkhitaryan signed a four-year deal with United in the summer after joining from German club Borussia Dortmund.
The Armenia international was named the Bundesliga player of the season in 2015/16 and scored 23 goals in all competitions for Dortmund.
Meanwhile, Liverpool are sweating over the fitness of midfielder Georginio Wijnaldum, who suffered a hamstring injury in the Netherlands' World Cup qualifying defeat against France.
The Reds hold a three-point lead over United heading into the crunch clash against their arch-rivals at Anfield.