The right-back is looking for a new club after being released by PSG at the end of last season.
Alves is one of the most decorated footballers in history, winning nine league titles with three different clubs, as well as three Champions Leagues with Barcelona.
And despite edging towards the end of his career, the 36-year-old shone for Brazil during their Copa America triumph this summer.
Alves has been linked with a switch to Manchester City, where he would be reunited with former Barca boss Pep Guardiola.
But according to beIN Sports reporter Tancredi Palmeri, Arsenal have entered the race.
“Arsenal in contacts with Dani Alves,” he tweeted.
The Gunners are desperate for fresh faces as they look to bounce back from a disappointing campaign.
Unai Emery has been handed a transfer budget of just £45m, with any additional funds expected to be raised by player sales.
“Arsenal in contacts with Dani Alves.”
beIN Sports reporter Tancredi Palmeri
Arsenal will need to be thrifty in order to improve their leaky defence and Alves on a free could represent good value.
Cover at right-back is needed, with Hector Bellerin still recovering from the cruciate ligament injury that forced him to miss the second half of last season.
Stephan Lichtsteiner has been released while Carl Jenkinson has been interesting the likes of Celtic and West Brom.
Alves played 32 times for PSG last season and is determined to continue at the highest level.
City are in the market for competition for Kyle Walker, with Danilo expected to leave this summer.
The Premier League champions had been linked with Juventus star Joao Cancelo but Barcelona appear to be leading the race for the Portuguese.
According to journalist Duncan Castles, Tottenham have also made contact about signing Alves.
Kieran Trippier is currently first choice at Spurs but there have been suggestions that the England international could move on.