The Daily Mail says yesterday the Spanish club made an opening offer understood to be around £72million.
The outline bid is still short of Liverpool's £80m valuation but shows an acceptance of the 27-year-old's worth.
Wednesday's meeting in London was productive and cordial with sensible expectations placed on both sides.
Although nothing was finalised there is an expectation that an agreement can be met to grant Suarez his wish to move to Barcelona.
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