At the bare minimum, the New York Yankees and Los Angeles Dodgers are expected to post the $20 million dollar maximum needed to negotiate with Tanaka. Other clubs are also expected to be in the mix. The New York Mets however, are not expected to be in that group per a report by Jared Diamond of the Wall Street Journal.
Projections have varied, but Tanaka is expected to receive contract offers exceeding $100 million dollars despite having never thrown a pitch in the majors. Even if the Mets had the financial capabilities to bid on Tanaka, I expect this is not the type of risk Sandy Alderson would prefer to take, so this news shouldn't surprise anyone.