The Council's branches reopened on Monday despite a government ban.
Russia called the move a "provocation" and said it would not issue new visas to British Council staff.
The offices were ordered to shut last month in the latest clash over the fatal poisoning of former Russian agent Alexander Litvinenko in London.
The branches which reopened are in St Petersburg and Yekaterinburg. The ban does not apply to the British Council's Moscow office.
A statement on the Russian foreign ministry's website said: "Russia views such actions as an intentional provocation aimed at inflaming tensions in Russian-British relations.
"The Russian side will not issue visas to new employees sent to work in the (British) consular offices of St Petersburg and Yekaterinburg to carry out British Council work."
The UK embassy in the Russian capital declined to comment on the schedule of the ambassador, Sir Tony Brenton.
But BBC Moscow correspondent Richard Galpin says the envoy will appear at the Russian foreign ministry on Monday afternoon.
He says the escalating diplomatic crisis between the two countries is now coming to a head and it is not clear if either side is willing to back down.
The British Council, which aims to promote cultural and educational ties, was accused of violating Russian tax rules.
A spokesman for the Council insisted on Monday that its activities were "fully compliant with Russian and international law".
"We believe we're caught up in a political matter," the spokesman told the BBC News website.
The British Council's chief executive, Martin Davidson, said they would continue talking to the Russians in the hope of continuing their work.
He told the BBC: "We have to understand the issues, the problems they have with our work, but we believe [our work] is enormously valuable.
"We know ordinary Russian people believe it is of real value to them, as well of course to us here in the UK, to have that relationship with Russia."
The offices in Yekaterinburg are on the premises of the UK consulate-general.
The row came on top of continuing tensions over the death of former KGB officer Alexander Litvinenko in London in November 2006. He had been given a fatal dose of radioactive polonium 210.
The UK wants Russia to hand over businessman Andrei Lugovoi, whom UK investigators suspect of murdering Mr Litvinenko.
When Russia refused to extradite Mr Lugovoi, Britain expelled four Russian diplomats and Moscow followed suit.
Russian officials have described the action against the British Council as a retaliatory measure.