The 31-year-old was recently left out of Thomas Tuchel's squad for their 1-0 Premier League victory away at Everton on Saturday.
Tuchel confirmed after the match that Alonso requested to be left out of the squad for the trip to Goodison Park and has agreed terms with a new club, although the German boss refused to reveal the defender's next destination.
"That is the case, he tries to leave and we agreed to this switch, and that's why it would have never made sense to have him on the pitch today. It's the right thing to do," Tuchel told reporters.
Spanish news outlet Sport have since posted a video on Twitter of Alonso's agent Fali Ramdani entering the office of Barcelona director of football Mateu Alemany to hold talks on Monday afternoon.
It is understood that Barcelona have reached an agreement with Chelsea to sign Alonso for a fee less than €10m (£8.4m), with the left-back agreeing a two-year contract.
There had been suggestions that Chelsea could use Alonso as part of a deal to recruit midfielder Frenkie de Jong from the Catalan giants, but Tuchel has recently played down rumours of a potential swoop for the Dutchman.
"Yeah, if we can add, we add. Maybe not necessarily a midfielder as a first option but we're open for everything," Tuchel told Sky Sports News on Saturday.
"I think we can use new energy, new legs and fresh input to bring us to a new level. If not, then we will push this team but I think we could use some more input and some more quality. We're on it but we will not tell you [who]."
Alonso is already understood to have said his goodbyes to the Chelsea staff at their Cobham training complex and his switch to Camp Nou could be completed at some stage this week.
Since joining Chelsea from Fiorentina in the summer of 2016, Alonso has recorded 29 goals and 23 assists in 211 appearances for the West Londoners across all competitions.
The nine-cap Spain international has also won a total of six trophies during his time at Stamford Bridge, including the Premier League, Champions League and Europa League.
The former Real Madrid youngster is set to become Barcelona's sixth signing of the summer window after the arrivals of Andreas Christensen, Franck Kessie, Raphinha, Robert Lewandowski and Jules Kounde.
However, it remains to be seen whether Alonso, along with Barca's other five new recruits, can be registered due to their ongoing financial problems.
The Catalan giants are required to significantly lower their wage bill before La Liga can allow the club to register new signings for the new campaign.
Both Martin Braithwaite and Samuel Umtiti have been told by president Joan Laporta to leave the club this summer, while goalkeeper Neto has recently completed a free transfer to Bournemouth.