Ursula von der Leyen, president of the European Commission, has said that if Covid vaccine supplies in Europe do not improve, the EU "will reflect whether exports to countries who have higher vaccination rates than us are still proportionate".
Post-Brexit disagreements between the EU and the UK have been heightened by the diplomatic row over the export of the vaccines.
The European Council president, Charles Michel, claimed last week that the UK had imposed an "outright ban" on the export of vaccines and their components - there is no ban though, and his claim was dismissed by the government as "completely false".
But Mrs von der Leyen says the EU is still waiting for exports from the UK, and it wants reciprocity.
How much vaccine has the EU been exporting?
The issue of vaccine exports from the EU - and a potential ban - is being raised because the EU is struggling to get sufficient supplies to accelerate its own vaccination programme.
And the number one export destination for vaccines manufactured in the EU is the UK.
Mrs von der Leyen says 41 million vaccine doses have been exported from the EU to 33 countries in six weeks.
More than 10 million of them have gone to the UK. That is more than the total number of vaccines administered in the UK in the month of February, and (as of 17 March) more than a third of the total number of UK vaccinations so far.
It is worth emphasising that vaccine exports are not organised by the EU itself, but by companies such as Pfizer and AstraZeneca, which use its territory as a global manufacturing base.
As of 11 March, 3.9 million doses had also been exported from the EU to Canada, and 3.1 million to Mexico. One million doses have been sent to the US, even though it is a major manufacturer in its own right and has not exported any vaccines to the EU.
The US is using export controls under the Defense Production Act, first introduced during the Korean War in the 1950s, to prevent companies exporting vaccine doses or ingredients without federal government authorisation.
What about UK exports to the EU?
There has been no public announcement of any vaccine exports from the UK, and no evidence that any have taken place.
The Department of Health said it didn't know whether there had been any, and AstraZeneca did not respond to a request for a comment.
"Let me be clear, we have not blocked the export of a single Covid-19 vaccine or vaccine components," Prime Minister Boris Johnson told the House of Commons on 8 March.
The government is keen to highlight that the UK has donated £548m to the Covax initiative, set up to distribute vaccines around the world. But that doesn't mean there have been exports of vaccines themselves.
"The British prime minister has made it clear to me that obviously his first priority is to vaccinate his people," Irish Prime Minister Micheál Martin said on 9 March.
"Until then he won't be in a position to give vaccines to anybody, and he has made that point to me."
No official ban
So, there is no export ban, but publicly available information suggests vaccines are not being exported from the UK. The government argues that is driven by the contractual obligations which vaccine suppliers have to their customers, rather than by the demands of politicians.
In January, the head of AstraZeneca, Pascal Soriot said of his company's contract with the UK that it was a case of "you supply us first".
And after the UK rejected Charles Michel's claim of an outright ban, he said there were "different ways of imposing bans or restrictions on vaccines/medicines". In an interview with news website Politico, he challenged the UK to release its vaccine export data.
Now, Mrs von der Leyen has stepped up the warnings.
"If the situation does not change, we will have to reflect on how to make exports to vaccine-producing countries dependent on their level of openness," she said.
In response, the UK Health Secretary Matt Hancock said the government had legally signed a contract for the delivery of the first 100 million doses of the Oxford-AstraZeneca vaccine, and added that "the supply of vaccines from EU production facilities to the UK is fulfilling contractual responsibilities and we fully expect those contracts to be delivered on".
EU's vaccine rollout
The EU has faced a series of problems with its vaccine rollout and has controls on exports, requiring manufacturers to seek permission from national governments for planned sales.
Earlier this month, Italy blocked a shipment of 250,000 doses of Oxford-AstraZeneca vaccine to Australia. But it is the only one of more than 300 vaccine export authorisations that has been refused.
The crunch could come in the second quarter of 2021 when supply problems may intensify. Then, as Mrs von der Leyen indicated, the EU may have to decide whether to block other shipments, including to the UK, to protect its own interests.
One possibility being discussed is to use Article 122 of the EU treaty, which allows measures to be taken "if severe difficulties arise in the supply of certain products". Those measures could in theory include export bans and the waiving of patent and intellectual property rights on vaccines.