The France international is out of contract at Stamford Bridge at the end of the season and there is little chance he will be offered a new deal.
The former Arsenal and Montpellier man was told he could leave in the January transfer window but despite links with Inter Milan, Tottenham, LA Galaxy and a few clubs in France's Ligue 1, the club eventually opted against a sale.
Giroud featured as a second-half substitute on Monday evening as Chelsea went down 2-0 at home to Manchester United in his first appearance since the end of November.
The 33-year-old has vowed to give his all for Chelsea between now and the end of the season but seems certain to depart once his current deal runs down on June 30.
France's Le 10 Sport has now reported that Lazio have made an offer to Giroud ahead of the summer transfer window.