The Merseysiders have been in talks about Can, a Germany Under-21 international, for some time and they have now made their move.
"Until Tuesday, it was a rumour," Leverkusen chief executive Michael Schade told German newspaper Bild.
"Now I can confirm that an official offer from Liverpool is received by mail."
Can has a 12million euro (£9.75million) release clause effective for foreign clubs and it is reported that the buy-back option Bayern Munich inserted into the deal which took him to Leverkusen for £4.4million last summer is no longer valid.
Liverpool were hopeful of progressing the deal last weekend and, despite claims Can's wage demands were a stumbling block, it is believed there were other as yet unspecified issues which needed to be sorted out.
If the deal is concluded Can would become Brendan Rodgers' second signing of the summer after he brought in striker Rickie Lambert from Southampton for £4million on Monday.
A take-it-or-leave-it offer of £25million for Lambert's team-mate Adam Lallana remains on the table but Liverpool are looking at alternatives should the Saints refuse to lower their £30million asking price, compounded by the 25 per cent sell-on clause agreed with Bournemouth.
The club are also trying to push through a deal for Sevilla left-back Alberto Moreno but are wary the Spanish club are trying to create an auction for the 21-year-old in order to inflate the asking price, which Liverpool currently believe is about £16million.