The Reds travel to Germany on Thursday for their last 32 clash, having managed to qualify for the latter stages earlier in the campaign.
It is unknown what sort of team manager Jurgen Klopp will pick though, having gone with a series of younger sides in their recent run in the FA Cup.
That strategy was largely enforced after injuries ravaged his squad, but with a bigger complement of players now available to him, Thompson says he would like to see a strong line-up in Germany.
"I would like to see Liverpool go for the Europa League,” he told Sky Sports.
“With a top-four finish looking a tall order, I also wanted to see them go for the FA Cup.
"Klopp can go with a strong team for the first leg as there's no game at the weekend, but it will be interesting to see how he prioritises the second leg, with the Capital One Cup final on the Sunday."
Klopp’s selection for the clash with Augsburg will likely be an indication of how he will approach the Europa League this season.
A young and inexperienced side would be a strong suggestion that he will be focusing on the Premier League at the end of the campaign.
On the other hand, a strong team will show he is serious about being successful and making a proper run at the tournament.
The latter would be the wiser choice, especially given Liverpool are unlikely to make the top four and that the winners of the Europa League are handed a place in the Champions League.