Dalot has allegedly emerged as the team's first-choice option on the right side of the defence due to his recent impressive form for the 20-time English champions.
The 23-year-old has been Man United's number one right-back since the end of last year, and he has again been a key player under Erik ten Hag, making 11 appearances in all competitions, contributing two assists.
Dalot's contract at Old Trafford is due to expire next June, and although the Red Devils have the option to extend his deal for a further 12 months, there has been speculation surrounding his future.
A recent report claimed that Barcelona had been in contact with the defender's representatives to discuss a possible switch to Camp Nou in 2023.
According to Sport, the Catalan outfit are making progress on a possible deal for Dalot, who is said to be open to making the move to the club despite becoming an important player for Man United.
The report claims that Barcelona are also stepping up their efforts to sign Martinez from Real Sociedad amid claims that Gerard Pique is likely to depart next year.
Martinez was heavily linked with a switch to Camp Nou over the summer, but the transfer window closed with the Spain international still at his Basque club.
The 31-year-old's existing deal is due to expire at the end of June, meaning that he would be free to hold talks with foreign clubs at the start of 2023.
However, Barcelona would have to agree a transfer fee with Real Sociedad if they wanted to bring him to the club during the January transfer window.
The Spaniard joined Athletic from Real Sociedad in January 2018, and he has made 163 appearances for the club, contributing seven goals and seven assists in the process.
Martinez has been restricted to just four La Liga appearances this term, though, as he missed the start of the campaign due to a knee problem.
There is thought to be a €80m (£68m) release clause in his current deal with the Lions, but they are not in a strong position due to the length of time left on his contract.
Barcelona strengthened their defence over the summer with the signings of Jules Kounde, Andreas Christensen and Hector Bellerin, but all three players are currently on the sidelines through injury.
Ronald Araujo is also out of action following thigh surgery, leaving head coach Xavi short of defensive options during a key stage of the campaign.