According to The Guardian, the prime minister will look to use his "improved relations" with the EU to secure the agreement.
The issue is said to have been raised by British diplomats, with a potential discussion tipped to take place between Sunak and European Commission chief Ursula von der Leyen in Japan in June.
If passed, the deal could help ease passport queues at airports, but concerns over the impact it will have on cross-Channel travel remain.
French police at the Port of Dover and at Eurotunnel in Folkestone will still need to physically check stamps on passports.
