Pogba was the target of racial slurs on Twitter when his penalty was saved by Wolves goalkeeper Rui Patricio in Monday’s 1-1 draw.
Then United striker Rashford suffered racist abuse online after hitting the post with a penalty in Saturday’s shock 2-1 defeat against Crystal Palace.
Chelsea striker Tammy Abraham and Reading’s Yakou Meite were also recently targeted by social media trolled and World Cup winner Pogba has had enough.
The 26-year-old insists he will use the jibes as fuel to step up the fight against racists.
“My ancestors and my parents suffered for my generation to be free today, to work, to take the bus, to play football,” Pogba wrote on his Instagram account on Sunday.
“Racist insults are ignorance and can only make me stronger and motivate me to fight for the next generation.”
During the Palace match at Old Trafford, United fans unveiled a banner reading ‘United against racism’.
Rashford had tweeted in support of Pogba after the Wolves games, saying: “Enough now, this needs to stop Twitter. Manchester United is a family. Paul Pogba is a huge part of that family. You attack him you attack us all.”
United manager Ole Gunnar Solskjaer condemned both incidents involving his players.
The Premier League club is believed to be planning talks with Twitter chiefs in a bid to resolve the problem.
Anti-discrimination campaigners Kick It Out have also called for “decisive action”.