Manchester United have agreed to sell Angel Di Maria to Paris St Germain for 63million euros.
Ronaldinho, 34, is available on a free transfer after leaving Atletico Mineiro in July and has reportedly attracted widespread interest.
Basingstoke marketing director Simon Hood says they have joined the chase for Ronaldinho and are waiting to hear whether the player is interested in a move to the Soccer AM Stadium.
“Discussions have been had, and there is an offer on the table. Now it is up to Ronaldinho to decide whether he wants to take us up on it," said Hood.
“He is on a free transfer and if he wants to get into English football, I cannot think of a better way to do that than by signing for Basingstoke Town.”
Chennai Titans are also understood to be keen on Ronaldinho and the player's brother, Roberto de Assis, is not ruling out the possibility of playing in India.
He told Globo Esporte: "We have been invited to this new league project, it's a short term project.
“There is no definitive answer yet but we liked what we heard."
Speaking about other offers, De Assis added: "There is a lot coming to a conclusion in the market. We are hearing everyone and will have news soon."