Goals in either half from Marko Grujic and Alberto Moreno, who converted a late penalty, gave them a fourth successive pre-season win on home soil against their Championship opponents.
But speaking afterwards, Klopp was not drawn on the futures of Flanagan and Allen - with the two on the verge of moves to Burnley, on a season-long loan, and to Stoke City, for around £13 million, respectively.
He told reporters that there is "nothing [to say] about this and all the transfer rumours" before insisting he "cannot comment" because otherwise they "couldn't do anything else all day if we did."
The Reds boss revealed: "I feel good [about Joe]. I like the guy and I really think he is a wonderful football player."
He insisted that he knew so even before the European Championships, as part of Chris Coleman's Wales side that reached the semi-finals, adding: "I saw it during the Euros and I know it now."
He declared that he has said "all I can say about this" and told reporters that it is not "all about respecting contracts, negotiating or not" or "making a transfer or not."
But he added that it is "nothing to discuss in public" because he has not seen the Welshman since "the middle of May in Basel" when the Reds lost to Sevilla in the Europa League final.
Flanagan and Allen on edge of exits as Reds boss trims numbers for 2016-17 campaign
Allen, whose potential new suitors included Sevilla until Stoke stepped up with a £13 million bid on Wednesday, is closing in on a move to the Britannia Stadium with his future at Anfield having long looked uncertain.
He recently entered the final year of his Liverpool contract and with no sign of a new deal, Allen is likely to move on in search of more regular first-team football.
Flanagan, meanwhile, was left out of the squad for Wednesday's win at Huddersfield's John Smith's Stadium to weigh up a potential season-long loan to Premier League new boys Burnley.
Despite strongly denying links to Celtic earlier in the week, both temporary and permanent, Flanagan could seal a temporary switch to the Clarets in order to continue his development.
He hasn't played regularly in the Premier League since 2013-14, due to injuries and competition for places on Merseyside, with Klopp seeing Burnley as the right place for Flanagan to get back into regular senior action.
Otherwise, Liverpool will also send Lucas Leiva for scans on the hamstring injury he suffered after an hour at Huddersfield.
The Brazilian midfielder was due to be in the squad that will travel to the USA for a 12-day tour on Thursday morning, although his injury could affect his involvement.
But having suggested only two of the youngsters who have impressed in the Reds' first four friendlies would travel, Ryan Kent is set to remain in England with a loan move to the Championship for the coming season looking likely.