Liverpool manager Kenny Dalglish says he has some bridges to rebuild in restoring the damaged confidence and reputations of some of the experienced members of his squad.
Konchesky has not played once in Dalglish's three matches in charge since taking over a little more than a week ago and was an unused substitute for Sunday's 2-2 draw in the 215th Merseyside derby against Everton at Anfield.
The 29-year-old was signed from Fulham on transfer deadline day in August but has failed to win over fans who believe he is not up to the quality required.
With the right-sided Glen Johnson filling in on the left for the last two matches and Liverpool being linked with a loan move for Aston Villa's former Reds defender Stephen Warnock - as well as a move for Ajax forward Luis Suarez - it appears Konchesky's days are numbered.
Exactly the same can be said of Poulsen, who will be 31 in six weeks' time.
He made only his first Barclays Premier League appearance in two months in Wednesday's defeat at Blackpool and was omitted from the squad yesterday as 22-year-old Jay Spearing made a surprise start in central midfield following two months out with a broken ankle.
Dalglish insists he has not written off their chances but prospects look limited for the pair.
"We've got a couple of senior players who the supporters haven't exactly seen eye to eye with," Dalglish said. "We've got to try to rebuild them. If we can get Christian Poulsen and Paul Konchesky back on side then that makes the squad that bit deeper.
"Christian played very well at Blackpool and Paul will get an opportunity as well.''
Of his decision to continue playing Johnson at left-back despite Konchesky and Fabio Aurelio being fit, he added: "We want our best players on the pitch. I don't know if he's the best option but he's an option."
After back-to-back defeats having taken over from Hodgson, Dalglish earned his first point in the entertaining derby - his first match at Anfield for nearly 20 years as Reds boss.
The Scot, who once again praised his players' character, believes performances are improving after Raul Meireles' first goal since signing for the club in the summer was overhauled immediately after half-time by efforts from Sylvain Distin and Jermaine Beckford.
"There is a lot we still have to analyse but really there has been a lot of positive things from the players," he said. "We were happy with the way we are playing. There is a lot of belief coming back into the players and that is great for us."