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.
"It's the same we spoke about before the weekend. It's unheard of and we need it to stop, it needs to stop," Solskjaer said.
"I'm just lost for words if it keeps going. We keep having all these campaigns 'No to Racism' and it keeps hiding behind fake identities. It's just crazy that we talk about this in 2019."
Pogba's message was accompanied by a photograph showing him holding his baby alongside portraits of his late father Fassou Antoine and civil rights activist Martin Luther King,
Twitter said on Wednesday that they will meet with United, Kick It Out (anti-discrimination organisation within football) and any other civil society stakeholders to discuss the problem.
Twitter claim to have already permanently suspended a series of accounts as a result of the racist posts.
Anti-discrimination campaigners Kick It Out have also called for action.
"The vile racist abuse on social media continued today. This problem will not go away and needs decisive action - that's what we'll be making clear to Twitter when we meet," Kick It Out said on Saturday.
"Without immediate and the strongest possible action, these cowardly acts will continue to grow."