The exchange occurred after the Irish mixed martial artist beat Donald Cerrone in his comeback fight in January.
McGregor, 31, shared the messages to prove he had been actively seeking fights before his latest retirement from the UFC, in June.
He made the decision after plans for another fight were postponed due to the Covid-19 pandemic.
This week, McGregor claimed his next fight would be against boxing legend Manny Pacquiao. The 41-year-old, now a senator in the Philippines, subsequently confirmed the news through a statement from his office.
Pacquiao said he would donate the majority of funds from the proposed fight to the Covid-19 relief effort in his native country.
“I had people blowing me up left and right,” White said in response to the flurry of viral tweets from McGregor.
“I mean everybody here knows, I mean even the ladies, this is some man-code stuff but it’s just something you don’t do. It’s just something that you don’t do, it is one of the dirtiest things you can do.
“Which, by the way, because we were just talking about Diego Sanchez. And Diego Sanchez was in there in this private conversation that I was having with Conor McGregor.
“When you are the number 2 or 3 ranked guy in the world, and you’re telling me that you want to fight unranked 39-year-old Diego Sanchez in the main event of Los Angeles. Well…”
Despite claiming to be retired from UFC, McGregor has claimed he is still interested in fighting Sanchez.
He accused White of 'lying', claiming he had been involved in talks about the potential fight against Pacquiao.
Taking to Twitter again, McGregor wrote: "Code was broke [sic] when you lied about me turning down fights mate. I said Justin in May and you went and said I did not want to fight. It’s not about Diego. Diego was a filler to get more fights in. Also you have been involved in Manny talks the legal letters are there. Stop lying."