Keane has fostered a strong reputation over his three seasons with the Clarets so far, and earned his first England cap against Germany in March.
The 24-year-old is set to be one of the most in-demand players on the Premier League radar this summer, with the Reds among those touted with a big-money bid.
And with Jurgen Klopp in need of at least one more centre-back ahead of the 2017/18 campaign, Liverpool are said to have made progress in their efforts to sign Keane.
According to ESPN‘s United correspondent Rob Dawson, Klopp’s side have “the upper hand” over their northwest rivals.
This is due to Keane’s “unhappy” departure from Old Trafford back in 2015, due to Louis van Gaal’s poor treatment, and he would need “convincing” to return to United.
Keane would be far from a blockbuster signing like Van Dijk, but he would certainly bring qualities that could improve the Reds’ defence.
He is a strong defender on a fundamental level, but is also comfortable with the ball at his feet, and looks capable of making the step up to the next level.
Burnley are believed to be seeking a fee of around £20 million for Keane, and if Liverpool were to sign both him and Van Dijk, this would mean an outlay of £70 million on two centre-backs.
This would be a bold move, but one that is necessary, and would be offset by a hopeful £30 million departure for Mamadou Sakho.