According to Sport Witness, both clubs were impressed by the Danish winger’s exploits at Euro 2020 and believe he would improve their attacking options.
Aston Villa have also expressed interest in Damsgaard over recent weeks as they adjust to life after Jack Grealish.
The loss of their captain and main creative outlet was mitigated, to a certain extent, by the signings of Norwich City’s Emi Buendia and Bayer Leverkusen’s Leon Bailey.
Damsgaard remains a possibility for Villa, but Liverpool and Leeds are believed to have a more pressing need for reinforcements.
The Reds continue to rely heavily on Roberto Firmino, Mohamed Salah and Sadio Mane, so would welcome another high-quality alternative.
Diogo Jota has chipped in with valuable goals since arriving from Wolverhampton Wanderers last year and will be needed more than ever while Salah and Mane compete in January’s African Cup of Nations.
Marcelo Bielsa's Leeds haven’t been at their fluent best so far, picking up just two points from their opening three games of the season.
Bringing in Damsgaard would certainly make a difference, with his skill, vision and eye for goal making him one of the breakthrough stars of this summer’s Euros.
The 21-year-old started the tournament on the bench but was drafted into the starting line-up from the second game onwards and soon made himself indispensable.
He scored a thumping long-range effort in the 4-1 win over Russia that secured Denmark’s progress to the knockout stages and registered an assist in the second round against Wales.
In the semi-final, his well-struck free kick beat Jordan Pickford to open the scoring before England fought back to win in extra time.
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