The Red Devils were knocked out of Europe earlier in the month by Barcelona and, with four league matches remaining, find themselves sixth in the Premier League.
Manchester United will need help from Arsenal, Chelsea and Tottenham to finish in the top four, particularly after Sundays woeful 4-0 defeat at Everton.
Missing out on Champions League qualification will have major consequences, with The Guardian saying a number of senior players would then consider transfer moves from Old Trafford.
It is claimed goalkeeper De Gea, midfielder Pogba and striker Lukaku would all contemplate summer departures if United missed out on a top-four finish this term.
Talks over a contract extension for De Gea have stalled and the Spaniard is believed to be interested in a move as he has aspirations of winning multiple trophies.
Pogba was heavily linked with a move to Barcelona last summer and returning Real Madrid manager Zinedine Zidane has expressed his admiration for the Frenchman this month.
Lukaku, meanwhile, has lost his place in Uniteds starting line-up to Marcus Rashford and The Guardian say an absence of Champions League football could be the final straw for the Belgian.