The 23-year-old has made a major impression with Ruben Amorim's side from the middle of the park and was a key figure in their 2020-21 Primeira Liga title triumph.
Nunes continued his fine form into the 2021-22 season, scoring four goals and setting up five more in 50 appearances across all competitions at club level.
The Portugal international has caught the eye of a number of clubs across Europe in the past couple of years, but he only extended his contract with Sporting until 2026 last year.
Wolverhampton Wanderers and Newcastle United have nevertheless been linked with a move for Nunes, who would likely command in excess of £50m to leave Sporting this summer.
Now, journalist Rudy Galetti reports that Liverpool and Chelsea are also in the race to sign the midfielder, but the duo will also face competition from Manchester City.
Galetti adds that Sporting are yet to receive an official bid for Nunes, and the race is open to prise the Portuguese away from the Primeira Liga giants in the coming weeks.
Nunes spent time on the books at Ericeirense and Estoril before arriving at Sporting in January 2019, and he has since racked up seven goals and eight assists in 99 matches for the club.
As well as scooping the 2021-22 top-flight title, Nunes has also won two Portuguese League Cups and one Portuguese Super Cup during his time with Amorim's side.
Liverpool are known to be in the market for a new midfielder, but Jurgen Klopp is expected to hold off on making a big-money move for a new lynchpin until the summer of 2023.
The Reds have tied James Milner down to a new one-year deal and have already splashed the cash to sign Calvin Ramsay, Fabio Carvalho and Darwin Nunez in the current window.
Borussia Dortmund's Jude Bellingham will likely be Liverpool's number one engine room target for next year, while City have already replaced Fernandinho with Kalvin Phillips.
However, Chelsea duo Jorginho and N'Golo Kante have both seen their futures called into question, as the pair will both be out of contract at the end of next season as things stand.
Kante has been linked with a shock move to Arsenal, but Thomas Tuchel would likely be reluctant to sanction either player's exit while his side focus on transfer business in the defensive and attacking thirds.