"I'm only coming out of retirement for one person and that's Conor McGregor.
"We don't need to waste no time. We need to make this ... happen quickly. Let's get it on in June."
McGregor has been challenging Mayweather, who retired in 2015 with a 49-0 record, to a fight for months.
The two sides have yet to reach terms and have a number of major hurdles to overcome.
Mayweather has asked for a guaranteed $100 million to make the fight happen, with $15 million going to McGregor, while UFC president Dana White has talked of $25 million to each plus pay-per-view money.
McGregor, the biggest pay-per-view draw in mixed martial arts, is contracted to UFC and the organisation would have to approve any fight.
The pair will also have to agree to a format which Mayweather insisted would be straight boxing.
"We cannot mix up the rules and regulations because I'm the A side," he said.
"He called me out. You call me out, then, we have to go by my rules.
"If I called you out then we'd have to go by your rules.
"It's always boxing."
The pair have done their part to drum up interest, exchanging a number of verbal barbs at public appearances.
Mayweather reiterated his claim that McGregor was all talk and no action.
"Conor McGregor, stop ducking and diving Floyd Mayweather. Tell the UFC to let's make this fight happen. I don't want the UFC to look bad."
(Reporting by Ian Ransom; Editing by Amlan Chakraborty)
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