Mr Hunt travels to Paris for a meeting with his French counterpart Jean-Yves Le Drian, before heading to Vienna on Wednesday for further Brexit talks where he will again warn of the possibility of an "accidental" exit from the EU without an agreement.
The Foreign Secretary will also meet the director general of the International Atomic Energy Agency, Yukiya Amano, to discuss the future of the Iran nuclear deal.
Mr Hunt said: "The UK and France have a partnership like no other, built through shared history, shared values and shared cultures.
"In Austria, the UK has a vital partner on trade, security and wider European issues for which together we work to defend the rules-based international order and safety of our citizens.
"The UK and its European friends are striving to build a new relationship that works for all of us. We have taken a principled and pragmatic approach through the white paper, and with our EU partners, we have made great strides towards that deep and special partnership.
"But it is vital this momentum continues, and I will be explaining to my counterparts in Paris and Vienna that it is time for the EU to engage with our proposals, or we potentially face the prospect of a no-deal by accident, which would be very challenging for both the UK and EU."
The visits follow a trip to China where Mr Hunt mistakenly told his hosts his Chinese wife is Japanese before correcting himself and admitting he had made a "terrible mistake".