With the future of Tottenham striker Jermain Defoe currently uncertain, the report claims Villas-Boas has identified the 13-goal Championship leading scorer as a potential replacement.
Defoe has been linked with a move to Toronto in the MLS, now managed by former-Tottenham defender Ryan Nelson.
The England striker has found himself largely restricted to substitute appearances since the summer arrival of Roberto Soldado, and Nelson is hoping to convince him to join him in Canada.
Earlier this month, Nelson said: "To get a guy like Defoe, one of the most natural goalscorers I have ever come across , for the MLS…I cannot think of many that are more suited.
"It would be incredible."
But with Spurs third-choice forward Emmanuel Adebayor also linked with a White Hart Lane exit in the January window, Villas-Boas would likely have to strengthen before he consented to Defoe's departure.
Ings, who's scored 13 goals for Burnley so far this season, is attracting Premier League attention having fired his club to the top of the Championship table.