The Spanish duo weren’t called into Vicente del Bosque’s Spain squad for the international break.
Moreno was sidelined with an injury for Liverpool’s 3-2 defeat by Southampton at St Mary’s on Sunday afternoon.
Enrique, meanwhile, has only made two FA Cup appearances since Klopp took over the reins from Brendan Rodgers last October.
However, the Liverpool full-backs looked in good form as they buckled up for their flight to sunny Tenerife for a warm weather training camp.
The Reds stormed into a 2-0 lead at St Mary’s in Moreno’s absence thanks to goals from Daniel Sturridge and Philippe Coutinho.
However, Sadio Mane netted twice and Graziano Pelle scored a curled effort to seal a 3-2 comeback victory for Ronald Koeman’s side.
Liverpool are in eighth place in the Premier League table.
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