According to British newspapers The Telegraph and The Times, the French are seeking Britain's senior position in Nato.
At the moment the UK holds the second most senior military position after the US, within Nato, as Deputy Supreme Allied Commander (DSACEUR)
The Telegraph points out that the reasons why questions are being raised, is because the position held by the UK is central in securing Nato manpower and equipment for EU missions as organised under the 'Berlin-Plus' agreement.
According to a report written by Professor Malcolm Chalmers for the UK based international relations and security think tank RUSI. "There is already discussion of the possibility that the position of Nato Deputy Supreme Allied Commander […] might be transferred to a Nato member that remains part of the EU."
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The Times reports that an unofficial French delegation went to Washington last year to convince US officials that its armed forces would be better placed to be the US's special ally in Europe post Brexit.
The French team "were at pains to point out how useful the French military could be as an ally and their track record in getting things done in trouble spots where the US was not strong," a source told the newspaper.
"They also pointed out that after Brexit, they would be the only EU country with this capability."
If the French were to get the position of DSACEUR, it would be a significant loss of prestige for the UK, which has held the role since 1951, when British WWII veteran Field Marshall Bernard Montgomery was appointed.
However Chalmers said "There may be creative ways to handle this issue, "such as creating a second DSACEUR position or the UK taking the role of Chief of Staff.
But he stressed "the fact that this is already being raised, is a clear message that the UK's influence within Nato, cannot be entirely ring-fenced from the consequences of Brexit."
He also warned the UK not to use "its role as a leading […] military and intelligence power – as a bargaining chip in the Brexit negotiations."
But Chalmers also pointed out, "The UK's contributions to European security can however help remind other EU states of the strong interests and values they will continue to have in common."
Even after Brexit the defence expert was confident that the UK's influence on European security "will remain considerable." But this will be hard to translate into "political influence."
"It will be hard to ensure that its policy inputs are not the afterthought of any US-EU dialogue."
Chalmers believes that following the UK's departure from the EU, it will no longer be represented at the hundreds of meetings in which the EU decides how to respond to international issues.
Nor was it realistic to "expect that the UK will be able to maintain a significant observer role."
Given currently there was no transition agreement planned, and with only two years of negotiations before the UK left the EU. Chalmers was worried that the government will be bogged down in talks concerning a range of issues, which as a result "there is a risk that foreign and security policy could be overlooked."
But Chalmers also stressed that despite the UK and the EU competing in many policy areas. There "will be broad geopolitical interests" on which they will be closely matched.
Therefore he called on the UK government to start thinking of a post-Brexit 'special relationship', with the EU on foreign and security policy.
"Such a relationship together with some strengthening of long-existing mechanisms […] could help to ensure that the UK will continue to be closely involved in consultations on the main security policy issues of the day."