The Magpies are understood to have come forward with an offer of £40m for Maddison, which Brendan Rodgers's side swiftly rebuffed.
Rather than try again for the playmaker - who is valued at £60m by Leicester and remains happy at the club - the Daily Mail claims that Newcastle will turn their attention to Barnes.
Enquiries have supposedly been made over a deal for the 24-year-old, and Leicester are under pressure to raise funds given their lack of incomings so far.
Barnes scored six goals and set up 10 more in 32 Premier League appearances last season and is under contract at the King Power Stadium until 2025.
Meanwhile, the future of former Arsenal and Juventus midfielder Aaron Ramsey remains shrouded in uncertainty after his contract with the Old Lady was terminated earlier this month.
The Wales international will be after regular game time ahead of the 2022 World Cup in Qatar, which has led to speculation surrounding a switch to MLS, where he would link up with compatriot Gareth Bale.
According to the Daily Mail, MLS side Charlotte Independence and Ligue 1 side Nice are the two contenders to sign Ramsey, who will hope to reach a final decision over his future this weekend.
The report claims that Charlotte are prepared to offer the 31-year-old a deal until 2024 which would see him become their highest-paid player.
Ramsey - who reached the Europa League final on loan with Rangers last term - left Juventus after registering six goals and six assist in 69 appearances since 2019.
Over at Elland Road, Leeds United are set to concede defeat in the race to sign Club Brugge attacker Charles de Ketelaere, who is on the verge of joining AC Milan.
The Belgium international is set to undergo his medical over the weekend, forcing Jesse Marsch's big spenders to seek out alternative options in attack.
A separate report from the Daily Mail claims that the Whites have now been offered the chance to sign Julian Quinones, who plies his trade in Mexico with Atlas.
The 25-year-old only signed for the club permanently from Tigres UNAL at the start of the summer after scoring 13 goals and setting up nine more in 42 games on loan last term.
Quinones is a centre-forward by trade, but the former Colombia Under-20 international is just as comfortable on the left or right wing.
While AC Milan are seemingly set to wrap up the signing of De Ketelaere without any major problems, their pursuit of Aston Villa's Carney Chukwuemeka is not going as swimmingly.
The 18-year-old is yet to agree to a new contract at Villa Park - with his current deal due to expire in 2023 - and was left out of Steven Gerrard's pre-season squad for their trip to Australia.
Milan are understood to be at the head of the queue to sign Chukwuemeka, but Italian outlet Gazzetta dello Sport - via the Birmingham Mail - claims that the Rossoneri are falling short of Villa's valuation.
The Lions are believed to be after a £20m fee to sanction a sale, but Milan are not afraid to hold firm on their stance in the hope that Villa will ultimately be forced to lower their asking price.
Chukwuemeka - who became European Under-19 champion with England earlier this year - has one assist to boast from 16 senior appearances for Aston Villa so far.