The 25-year-old is widely expected to leave the Foxes in the current market following their relegation from the Premier League, and a number of clubs have been linked with his services.
According to The Guardian, Aston Villa, Tottenham Hotspur and West Ham United are all showing an interest, but Newcastle are the team leading the race.
The report claims that Leicester valued Barnes at £50m earlier this year, but his price has now dropped following the club's relegation to the Championship.
The one-cap England forward enjoyed a relatively strong 2022-23 campaign for the Foxes despite the team's struggles, scoring 13 goals and registering one assist in 34 league appearances.
Inter Miami, meanwhile, are allegedly closing in on the signing of Jordi Alba on a free transfer.
The 34-year-old's deal at Barcelona had initially been due to expire in the summer of 2024, but an agreement was reached with the Catalan outfit for him to depart on a free transfer this year.
Alba's future is currently the subject of much speculation, with Atletico Madrid thought to be keen to keep him in La Liga, while a number of Saudi Arabian clubs have been linked with his services.
However, according to The Athletic, Inter Miami are now confident of securing a deal for the experienced defender, who is preparing to follow Lionel Messi and Sergio Busquets to the MLS outfit.
Alba made 459 appearances for Barcelona during a highly-successful period in the club's history, scoring 27 goals and contributing 99 assists in the process.
Wolverhampton Wanderers, meanwhile, have reportedly joined West Ham United and AC Milan in the race to sign Universidad de Chile attacker Dario Osorio this summer.
The 19-year-old is expected to leave his Chilean club during the current market, with a number of clubs said to have registered their interest in his services.
According to The Sun, Wolves are keen on the youngster, who has more than two years left to run on his current deal, placing Universidad de Chile in a strong negotiating position.
Osorio has scored nine goals in 44 appearances at club level, while he has been capped on three occasions at senior level by Chile, making his debut in a friendly last June.
Sheffield United, meanwhile, are reportedly interested in signing Nottingham Forest midfielder Lewis O'Brien for their return to the Premier League.
The 24-year-old only made the move to Forest from Huddersfield Town last summer, but he was sent on loan to DC United in March, with his agreement with the MLS outfit set to expire in the middle of this month.
DC United head coach Wayne Rooney has expressed a desire to keep hold of O'Brien, but according to Yorkshire Live, Sheffield United want to sign him on a permanent basis.
The report claims that Forest head coach Steve Cooper is yet to decide whether the midfielder will be in his plans for the 2023-24 Premier League campaign.
O'Brien made 17 appearances for the Reds in the first half of last season but was then left out of their 25-man squad for the second half of the campaign.