Gunners boss Arsene Wenger also recently confirmed that a swap deal involving Sanchez and Mkhitaryan was likely to be completed.
However, it had been previously claimed by the Armenian international's agent Mina Raiola that a United move for the Chilean forward hinged on whether Mkhitaryan would accept the switch to north London.
According to the Mirror, Mkhitaryan has now agreed terms for a move to the Emirates Stadium and is already in London for a medical to take place on Sunday, 21 January, to finalise his transfer.
Meanwhile, United will get their man in Sanchez without the addition of a transfer fee, which was previously expected in the swap deal.
The 29-year-old had already agreed a bumper four-and-a-half year deal worth at least £400,000 per week ($554,400 per week) that blew previous favourites and neighbouring rivals Manchester City out of the reckoning.
The same report understands that Sanchez, who missed Arsenal's 4-1 demolition of Crystal Palace, left the team hotel on Friday, 19 January, night to complete his medical in Manchester.
United manager Jose Mourinho confirmed after his side's 1-0 win at Burnley on Saturday that a deal for the Chile international is "so close" as he expects it to be completed soon.
"If you ask me if I think he is coming, I think so, but I have no confirmation," Mourinho said.
"Expect soon or never. So close, so close, so close, if it doesn't happen, it is not going to happen. I know my people are doing absolutely everything they can. I think they will be successful."
Sanchez will leave Arsenal with two FA Cup titles, having scored 80 goals in 165 games in all competitions in a three-and-a-half-year spell.
Mkhitaryan, meanwhile, will leave United after just a season-and-a-half, with the 28-year-old winning an EFL Cup and a Europa League title.