Rickie Lambert has passed a medical and is expected to complete his £4million move back to his boyhood club Liverpool on Monday, while Press Association Sport is reporting that manager Brendan Rodgers has made another push for midfielder Lallana.
The Reds had an initial bid of £20million rejected but have gone back to Southampton with an increased amount.
However, as with Lambert, the Merseysiders are not prepared to hang around in their pursuit of targets and the offer effectively amounts to take-it-or-leave-it.
Lallana is believed to be keen on a move to Anfield so it leaves Saints in a tricky position as if they reject Liverpool's advances once again they risk alienating a player who has been a loyal servant to the club.
Liverpool have adopted the same approach with Lallana, Rodgers' primary target this summer, as they have with Lambert.
The 32-year-old striker emerged as a surprise option for the Reds during their on-going discussions with the south-coast club over Lallana.
When it became apparent there was a deal to be done the Anfield club insisted it be completed quickly and set Southampton a deadline of this weekend - before the England squad fly out to Miami for the next stage of their World Cup preparations - on Sunday.
Lambert confirmed that he had undergone a medical and that everything would become clear on Monday - the day of which the deal should be finalised.
The striker told Sky Sports News: "The medical went well but as you know I can't say anything.
"I'll speak properly about it, hopefully, very shortly.
"It's been a crazy football year to be honest, the way it's carrying on."
Lambert was asked if he had spoken to Liverpool captain Steven Gerrard about the move while with him on England duty over the past week.
He said: "I've spoken to Stevie a while ago, but I'll speak properly on Monday."
Considering Southampton have yet to accept the offer for Lallana that may take some pushing to get it over the line quickly but national team boss Roy Hodgson has already stated he would like the future of players considering moves this summer to be sorted before the tournament in Brazil begins.
Should the £25million offer for Lallana not be accepted Rodgers will move on to other targets, with Bayern Munich's Xherdan Shaqiri an alternative, but there is a hope pressure from the player - who wants a move to Anfield - may still see the deal done.
Ciao Anfield tweet: According to Kicker, Liverpool have submitted an offer for Bayern Munich's Xherdan Shaqiri. #LFC
Lambert's move, which even with a number of add-ons will not get close to the £9million being suggested in some quarters, is more straightforward.
The player was keen to rejoin the club he left as a teenager and Saints found it difficult to stand in his way given the forward's service to the club.
Lambert was born in Kirkby, Merseyside, and was a member of Liverpool's academy until his mid-teens when he was released and found himself plying his trade in the lower leagues for the likes of Macclesfield, Stockport, Rochdale and Bristol Rovers before finding success at Southampton.
He will provide some much-needed back-up to frontline strikers Luis Suarez and Daniel Sturridge at Anfield and, while he may have to content himself with a supporting role from the bench at Anfield, the chance to rejoin his boyhood club and play in the Champions League was an opportunity he did not want to miss.
Another Southampton player, defender Dejan Lovren, remains a target but there are others, although Press Association Sport understands Liverpool's interest in Bayer Leverkusen midfielder Emre Can is running into complications.
The two clubs plan more talks over the weekend which are likely to give a clearer idea of whether the Germany Under-21 international will also move to Merseyside.
Although there is a buy-out clause of about £10million which Liverpool are willing to meet there is another stumbling block to be overcome.
Bayern Munich have first option on a buy-back of the player they sold last summer but it was believed player's wage demands were a cause for concern but another, as yet unspecified, issue has cropped up.
Liverpool have made a bid, reported to be in the region of £16million, for Sevilla left-back Alberto Moreno but are still in negotiations with the Europa League winners.
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