STRASBOURG (Reuters) - British authorities should be "under no illusion" on financial services in future relations with the European Union after the Brexit transition period ends, EU chief negotiator Michel Barnier said on Tuesday.

Britain and the EU will lose privileged access to their respective financial markets when the Brexit transitional period expires at the end of December. After that, financial firms could continue servicing clients on the other side of the Channel only in some sectors and where rules are deemed equivalent.

In a speech to the European Parliament in Strasbourg, Barnier made it clear that equivalence regimes, which govern relations with foreign partners on specific financial sectors like clearing houses or stock exchanges, would remain under tight EU control, with no special treatment for Britain.

"There will be no general, global, permanent equivalence" with Britain, Barnier said, adding that the EU will maintain control of this and would decide alone whether to grant or withdraw equivalences.

"There will be no common management," Barnier added.

Faced with a huge, rival financial centre on its doorstep, the EU is keen to maintain complete control on equivalence regimes which it can currently revoke at short notice and unilaterally.

Britain is instead pushing for a mechanism that would allow a sort of joint control on equivalences, in a bid to offer more certainty to financial firms which serve EU clients from their London base.

But spooked by the uncertainty of the future equivalence regimes, many banks, asset managers, insurers and trading platforms based in Britain have opened EU hubs to continue servicing their continental clients.

(Reporting by Marine Strauss and Francesco Guarascio; editing by Philip Blenkinsop and Ed Osmond)