Chelsea boss Jose Mourinho is exploring his options with a number of names, including former-Stamford Bridge striker Didier Drogba, among the contenders. Arsenal have also been linked with Lavezzi.
Chelsea's forwards struggled to live up to expectations last season, and Samuel Eto'o has not had his contract renewed. Fernando Torres and Demba Ba are not expected to play big roles next term, especially following the signing of Costa from Atletico Madrid.
But Mourinho wants to supplement the Costa addition with another move for a second forward player to play off the Spain international.
Real Sociedad's Antoine Griezmann has been linked, as has Paris St Germain's £18m-rated Lavezzi.
Lavezzi for club and country
The former-Napoli striker scored nine goals in 24 starts from PSG last season, as the French champions defended their league title.
But the Lavezzi, Edinson Cavani, Zlatan Ibrahimovic triumvirate has not worked as hoped in the Champions League for the big-spending Parisians, and Lavezzi is thought to be the most expendable.
PSG were linked with a move for Chelsea playmaker Eden Hazard, although that now looks dead in the water with Mourinho adamant he will not be sold.
Oscar is another mentioned as a target for the Ligue 1 side, and a deal could perhaps be worked out involving the Brazilian and Lavezzi.
While Oscar and Brazil exited the World Cup in humiliating fashion this week, Lavezzi and Argentina progressed to the final after a nail-biting penalty shoot-out victory over the Dutch.
Lavezzi has made five appearances for his country, including three starts, but he's yet to find the net and has registered just one assist as Argentina have relied heavily on talisman Lionel Messi.