Talks between Saint Maxinin's advisers and Al Ahli are at very early stages.
Newcastle manager Eddie Howe is keen to sign £35million-rated Barnes from Leicester, but expects strong competition from West Ham.
The Foxes are in full rebuild mode after their relegation from the Premier League with some of their prized assets being picked up.
James Maddison has already join Tottenham and Barnes could be next in line to go.
The 25-year-old scored 13 Premier League goals last season in what was a dismal campaign for Leicester.
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Barnes has been with the Foxes since he was nine years old, going on to make 187 appearances for the first team, via loans at MK Dons, Barnsley and West Brom.
For Newcastle, it has been a relatively quiet transfer window so far.
The only signing has been Sandro Tonali from AC Milan and manager Eddie Howe has admitted frustrations with how the window has gone.
"Yeah I think at different times there has been frustrations," he told The Chronicle.
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"There have been difficult days, because obviously you want the end result, you want the best squad you can get."