The Uruguay striker previously enjoyed four productive years with the Eredivisie giants between 2007 and 2011.
He plundered 111 goals in 158 appearances during that spell, tasting title success in his last full season.
In January of 2011, Suarez left Amsterdam for Liverpool and eventually moved on to Barcelona three-and-a-half years later.
The goals have continued to flow at Anfield and Camp Nou for a man who is now 32 years of age.
Suarez recently revealed that he would be happy to see out his playing days in Catalunya, but Ajax have made him aware that he would be welcome back in the Netherlands if a switch ever appealed.
The Dutch club’s sporting director, Marc Overmars, told VTV Uruguay: “Suarez can come back to Ajax.
“The doors are always open here. I hope they replace him!
“No, I'm joking, but if he was really interested in coming back to Ajax, I would go to Barcelona the next day to speak with him.
“I think one of Suarez's biggest qualities is his mentality. The power to win, to score goals... he is one of the best in the world.”
While Ajax are open to bringing in a star turn from Barcelona, they have already agreed to letting one of their own head in the opposite direction.
Frenkie de Jong has a €75 million (£65m/$85m) deal in place with the Liga champions and will complete a big-money switch at the end of the season.
Overmars, who represented the Blaugrana in his playing days, added on the highly-rated 21-year-old midfielder: “It's important for Barca to have players like De Jong. With his talent, at his age, he could become a Xavi or an Iniesta.
“[Barcelona] have to bring through their own players and mix them with good players from elsewhere.
“De Jong's not Messi, there's only one [Messi]. But he's a player that fits in really well at Barca. I'm excited to see him there next season.”
It may be that he is not the only current Ajax star turning out for Barcelona next season, with teenage defender Matthijs de Ligt another of the exciting academy graduates to be attracting interest from Camp Nou.