The incident happened at around 11pm on Tuesday, in Dalkeith, when a Ford Focus car parked in St Andrew’s Street was deliberately set on fire.
The vehicle was extensively damaged in the blaze.
Police are keen to trace a man who was seen running from the area at the time of the fire starting.
He is described as white, 5ft 10in tall with a slim build and wearing dark clothing and a dark beanie hat.
Police have appealed for any witnesses to come forward. A spokesman said: “No-one was injured during this incident, but the vehicle sustained significant fire damage and we are urging anyone who witnessed any suspicious activity in the area to contact police immediately.
“We are also keen to trace the male seen within the area around the time of the fire and would urge anyone who recognises his description to come forward.”
Anyone with information is asked to contact police on 0131-311 3131, or the charity Crimestoppers in complete anonymity on 0800 555 111.