The Germany Under-21 international joined from Bayer Leverkusen after the Reds triggered a £9.75 million release clause.
Rodgers has already strengthened his forward line and midfield and while he wants to add more creativity in the form of Lambert's Saints team-mate Adam Lallana, defence may be the next area to bolster.
A £25 million take-it-or-leave-it offer for Lallana remains on the table and while Liverpool are keen for the deal to be done before the start of the World Cup that looks increasingly unlikely and they are looking at alternatives.
Southampton's appointment of a new manager, which could happen as early as next week, may ease the impasse which currently exists and it could be Liverpool have more luck in securing Sevilla left-back Alberto Moreno.
Interest has been registered with the Europa League champions but the Merseysiders are wary of the Spanish club's attempts to generate an auction in order to push up his value, which Liverpool currently assess at about £16 million.
Reports of Tottenham entering the race with a better cash-plus-player offer were quickly dismissed which leaves the Reds still in pole position and with Moreno not going to Brazil with Spain it is a deal which could be finalised quicker than the pursuit of Lallana.
With the future of centre-back Daniel Agger in doubt Rodgers is also interested in Dejan Lovren, also at Southampton, but that may well be the last of the moves to be made as Rodgers would probably want Agger off the wage bill before bringing in a replacement.
Can's arrival, which will not be confirmed until July 1 because of the formalities of international paperwork although Press Association Sport understands he has already signed a contract, has at least been wrapped up fairly quickly.
Once the release fee was met there was the matter of a buy-back clause Bayern Munich inserted into the deal which took the 20-year-old to Leverkusen for £4.4 million 12 months ago and it is reported the Bavarian club will earn £1.2 million in compensation as a result.
In a short statement, Liverpool said: "Liverpool have today agreed a deal in principle with Bayer Leverkusen to sign 20-year-old Emre Can, subject to documentation being completed."
Leverkusen chief executive Michael Schade, in confirming Can's departure, said the club were powerless to keep the player once Liverpool had come up with the correct fee.
"Emre Can has developed very quickly at Bayer 04 Leverkusen over the past twelve months," he told the club's website.
"We wanted to keep Emre at Leverkusen but the release clause is part of his contract.
"So we wish him all the best and lots of success in his time in the Premier League."
Away from team affairs, Liverpool have taken another step towards the redevelopment of Anfield by submitting a formal planning application.
The club have sent extensive details of how they want to expand the Main and Anfield Road stands - extending the ground's capacity to almost 60,000 - to Liverpool City Council and a planning committee will consider the proposals later this year.
"A lot of good work has been done and we still have more work to achieve certainty; however, the planning application is another step in the right direction," Ian Ayre, who has been promoted to a more wide-ranging role as chief executive by owners Fenway Sports Group, told Liverpool's official website.
If planning consent is granted construction will begin on the Main Stand early next year and be complete in time for the 2016-17 season.
- Sports & Recreation
- Bayer Leverkusen
- Adam Lallana
- Rickie Lambert