The club currently has Italy's top goalscorer in Ciro Immobile, who scored 36 goals in the Serie A to equal the record held by Napoli's Gonzalo Higuain in the 2015/16 season.
However, reports suggest Lazio are looking to find a replacement for Immobile's striking partner Felipe Caicedo, who has been released and is reportedly heading to play in Qatar.
Lazio were reportedly initially interested in Bournemouth forward Joshua King, but the English club were asking for a €30 million ($53 million) transfer fee, which Lazio considered to be too high.
The Italian club has now reportedly set its sights on Wood, who is coming off his best season in the Premier League.
The 28-year-old scored 14 goals for Burnley this season to help the team to a 10th placed Premier League finish and took out the award for top scorer at the Lancashire club's virtual prize giving ceremony.
"It has been a great season for everybody," he said at the awards show.
"I am looking forward to the battle next season with the boys up top, it's always tough to finish on top and it always goes down to the wire."
Wood joined Burnley from Championship club Leeds United for £15m ($26 million), a club record at the time. However, the Kiwi striker will likely be valued higher now after several strong seasons in English's top flight.
Last November, he extended his contract with Burnley until 2023, saying: "It's nice to have the two years under my belt and to be settled and happy where I am and now it's about pushing forward and doing even better."
While Wood appears happy and settled at Burnley, the lure and prestige of Champions League football and playing at one of Italy's biggest clubs may be a drawcard for the striker.
A report from the Lancashire Live – a website that covers Burnely football club – suggested a "hefty offer" for Wood would be needed in order tempt the club.
"He is a key man at Turf Moor and [Burnley manager] Sean Dyche will not want to lose a man who signed a new deal until 2023 last November," the report said.
"The Clarets are not in a position where they need to sell their key players and would be loath to lose their £15 million joint club-record buy."
The new Serie A campaign is set to kick off on September 19, a week after the Premier League starts.