The Premier League giants are reportedly keen to sign the defender in January.
The Express claim that both of the Premier League sides sent scouts to watch the Saddlers' midweek defeat to Swindon Town.
Walsall manager Dean Smith suggested that while the club would struggle to turn down an astronomical offer, he is not entirely convinced that the 18-year-old wishes to leave the Bescot Stadium.
"It's going to take a hell of a lot of money to take Rico away from this football club," Smith claimed in the Daily Express.
"Every player has a price don't they? We all know that. But it would take a very big offer and I'm not sure he'd want to go, at the moment, as well," he added.
Henry has been a prominent part of the side that have surprised many by sitting fourth in the League One table. The club boast the second best defence in the division, and the youngster has also proved that he has a couple of goals in his locker.
He bagged a late winner to beat Championship high-flyers Brighton in the Capital One Cup in August, and scored the only goal during an away victory over Scunthorpe in September.
While there is little question that Henry would be nowhere near ready to hold down a first-team spot at either club should he move during the upcoming window, both sides are arguably in need of cover in that area of the pitch.
The Red Devils lost Luke Shaw to a long-term injury earlier in the season, while Jurgen Klopp's side have often used Joe Gomez - who is more comfortable playing in the centre or on the right of the back four - or Alberto Moreno. Should Liverpool sign Henry, it would allow Gomez to concentrate on his favoured positions, as he'd no longer be acting as cover for the Spaniard - once he returns from injury.
Given the almost immediate success that Dele Alli has enjoyed to his Premier League career, it would not be a surprise if either side took a punt at trying to find their own superstar currently lurking in League One.