Klose is out of contract at the end of June and has been linked with both the Biancocelesti and Milan as a free transfer.
"Let's say that I am evaluating some offers and Lazio are among them," Klose told Golmania.
"My wife and I have been to Rome on vacation and it is a fantastic city. The food is wonderful, too.
"I was close to Lazio in January, but Bayern wanted money that they weren't prepared to pay. Now I am in charge of my own destiny.
"I tracked their progress and they have a very solid squad including Mauro Zarate, Hernanes, Cristian Ledesma and Andre Dias. I also like the club colours, as there is a lot of joy around the club and that means a lot to me.
"I need to talk to the Lazio directors, but can tell the fans that if I do come, then I won't let you down! I am very enthusiastic, I work very hard and never give in. My shirt is always wet through with sweat at the final whistle."