The 21-year-old Spain international is considered to be high on the list of targets for manager Brendan Rodgers, and a formal offer has now been made.
It is also understood that the Reds’ 20-year-old Spanish midfielder Suso, who spent last season on loan at La Liga side Almeria, could be used as part of the deal.
Moreno, capped twice for Spain and part of their provisional 30-man squad for the upcoming World Cup, made 40 appearances in an eye-catching season as Sevilla finished fifth in La Liga.
He made his debut in early 2012 and 19 appearances over the subsequent 2012-13 season, but the most recent campaign has very much been considered his breakthrough at the Ramon Sanchez Pizjuan.
Having narrowly missed out on a first ever Premier League title this season, Liverpool boss Rodgers is keen to strengthen the depth of his squad ahead of a first UEFA Champions League campaign since 2009-10.
Having signed a new long-term deal, Rodgers said he is reluctant to use the loan market again, suggesting a good number of new permanent signings could be made.
“The players we wanted we couldn't get last year but we're hopeful that this summer will be different,” he said.
“Hopefully that will mean it will be more permanent players than loan players but I can never rule it out completely.
“We are really preparing for next season. I've had some very good meetings with our recruitment team and we are very much together as one."