Ole Gunnar Solskjaer's side see Saul as a good fit to partner Nemanja Matic and Paul Pogba in midfield after ESPN FC revealed last week that Ander Herrera is seriously considering a lucrative three-year offer to join Paris Saint-Germain when his contract at expires in the summer.
United scouts have been closely monitoring the development of Saul since his impressive loan spell at Rayo Vallecano in the 2013-14 campaign.
Old Trafford officials are aware about the difficulty of the negotiations with Atletico set to demand Saul's full €150 million release clause, part of a deal the 24-year-old midfielder signed in 2017 which commits him to Atletico until 2026.
Los Colchoneros have already sold defender Lucas Hernandez to Bayern Munich for €80m and are preparing for a busy summer transfer window with Diego Godin, Filipe Luis, Antoine Griezmann and Rodri facing uncertain futures at the club.
Godin is expected to complete a free agent move to Inter Milan, Filipe will also be out of contract at the end of the season and last month ESPN FC revealed that Griezmann has offered himself to Barcelona and a number of Europe's other big clubs following a frustrating season at the Wanda Metropolitano.
Meanwhile, sources have told ESPN FC that Atletico also face a huge battle to keep Rodri with City boss Pep Guardiola earmarking him as the ideal long-term replacement for Fernandinho.
City have also considered the signing of Saul as officials at the Etihad Stadium value his versatility to adapt to different roles in midfield and defence.
But sources have told ESPN FC that Guardiola believes Rodri would be more suited for the position as the 22-year-old is the real holding midfielder at Atletico Madrid with Saul playing in a more advanced role and even as a full-back and a centre-back.
Rodri has been likened to Guardiola's former key man at Barcelona Sergio Busquets since he made his La Liga debut with Villarreal in December 2015.
Barca were heavily linked with his services last summer as they looked for a long-term replacement for Busquets, but Atletico won the race after agreeing to pay Villarreal around €20m plus €5m in add-ons for his services.
City also monitored Rodri during his time at Villarreal and sources have told ESPN FC they have increased their scouting of him ahead of making a move in the summer.
Although Atletico want to keep Rodri they will be unable to stop City triggering his €70m release clause, which the Premier League side view as affordable for a player his age.