The 29-year-old striker is thought to be stalling on a new mega-contract worth a staggering £825,000-a-week.
United would be able to afford the Argentinian ace should he be available, but club chiefs believe the lure of Pep Guardiola would give City the edge in landing the superstar.
An Old Trafford source said: “City would be the favourites if he were to one day leave, but we would obviously love the sign a player of that level.
“We can certainly compete on the cash front and Messi is an Adidas player like Paul Pogba, so we have our pluses.
“But all the same, we would not get involved in any bidding war unless Messi’s camp gave an indication that he was willing to come here.”
It is believed Messi wants to stay and finish his career with Barcelona, and that he will sign a new contract in the near future.
But rumours of a dispute have put a number of clubs on alert including Manchester City, United, PSG and cash-rich sides in China.
The five-time Ballon d'Or winner is the next in-line of stars the Catalonians have been trying to tie-down to fresh deals.
Neymar and Javier Mascherano have penned contracts and Luis Suarez is closing in on terms as Barca fight off the Premier League's huge TV money.
Barcelona president Josep Maria Bartomeu said yesterday he was “convinced” Messi would stay at the club.
He added: “All of Barca believe this is how it has to be, because he is the player that in the last few years has brought us sporting success.
“I believe we've got a team with incredible players, all of them. There is nonetheless a leader of course, Leo Messi, he is well known and popular all over the world.
“We're all convinced, we're all positive Leo Messi will end his sporting career in Barca.”
Messi has scored an incredible 433 times in 456 appearances during his time at the Nou Camp.