The couple, of Bagshot, Surrey, who have Alzheimer’s, paid the men who pretended to work for Steve Goss, of Goss and Sons, in Bracknell.
Surrey Police is investigating after the victim’s bank tipped them off.
The BBC said Goss is now offering to do the work for free, despite police saying he is also a victim.
Goss said: “I’m very angry about this and hope to find out who this is that’s been using my company title.
“All they’ve done, apparently, is put two tarpaulins on this gentleman’s roof and dislodged two ridge tiles, which would amount to no more than a £200 job.”
Det Con Stuart Pittaway said: “This is something that’s happening more often, so we are asking people to be more vigilant.
“If people do come to the front door, check identification.
“If you’re not happy with the identification, ring up.”