The two biggest combat stars have been pursuing a lucrative super fight for most of the year, with negotiations on an agreement being the major roadblock as of now.
However, following UFC 211 in Dallas, Texas on Saturday (13 May), White said he is expecting an "executed agreement" on 14 May from the lightweight champion.
"We're right there – I'm expecting an executed agreement tomorrow," said White at the post-fight conference for UFC 211, as quoted on MMAJunkie. "And then I'll start working with Team Mayweather next week."
"We're just getting this deal done with Conor. What do you think the Mayweather side's going to be like, and how much time am I going to spend on this? We're going to give it a shot."
Despite another recent setback on the date of the fight, with Canelo Alvarez and Gennady Golovkin's eagerly awaited middleweight title showdown confirmed for the desired 17 September date, White is still eager to get the deal done.
A boxing match between the duo would definitely be the most lucrative event that he would have promoted, with McGregor claiming he would make $100m (£77.3m, €92m) from the bout. However, White has his limits, especially if Mayweather's demands are "ridiculous" along with the added risk of a pay-per-view event not living up to expectations.
"It depends on how ridiculous it is," he added. "The question is how profitable (it could be for us). You can assume how many pay-per-view buys you're going to do. There's no guarantee. Every time you put on a pay-per-view, it's a roll of the dice. You don't know what's going to happen.
"There's some fights that you do know, and this feels like one of those fights that should do that. But you never know."