The news indicates that an £88m summer move to the Catalan giants has been green-lit.
The France international opened the scoring as Atletico beat Celta Vigo 3-0 in La Liga earlier today.
After a slow start to the season Griezmann has been finding his form again recently.
Today's strike took his tally to 21 goals for the season in addition to 12 assists.
According to Spanish publication Marca, the striker and his wife Erika Choponera used Atletico's recent game against Barca to check out areas they have pinpointed.
Catalunya Radio claim the Griezmann family went to Castelldefels, a town 20km south of Barcelona where Leo Messi, Luis Suarez and Philippe Coutinho live.
Then he visited homes in the neighborhood of Pedralbes, an area within the centre of Barcelona.
Ousmane Dembele lives in the area and Neymar had a house in the same neighbourhood during his spell at the club.
Apparently Choponera did not find anything that ticked all her boxes so they moved on to a third location.
Diego Simeone, Fernando Torres, Cristiano Ronaldo and Karim Benzema live in an area known as La Finca.
She reportedly wants a property fully equipped with private security and sole access.
There are no such complexes within Barcelona itself so their search will continue further afield.
Barcelona despatched Malaga 2-0 yesterday, Suarez and Coutinho getting the goals before Samuel Garcia Sanchez saw red for the hosts.