The Yemen peace process is in the “last chance saloon” and could collapse within weeks unless both sides shift position and agree to implement the terms of a ceasefire agreed by the UN-backed government and Houthi rebels, Jeremy Hunt said on Sunday.

The foreign secretary said the key issue was over trusting the Saudi-backed Yemen government not to take over the strategic Red Sea port of Hodeida as soon as the Houthi rebels left the town and handed tpolicing to a locally run force.

Speaking to the Guardian from Dubai after becoming the first western foreign minister to visit Yemen since the war started in 2015, Hunt said he hoped he had persuaded both sides to shift, but admitted he did not expect anything to happen in the next few days.

Referring to the ceasefire agreed in Sweden in December, he said: “There is a real issue of trust between the two sides, but it has now been 80 days since the Stockholm agreement, so there is a real sense of urgency. I have no idea if me being out here had any impact or not, but the reason to come here was to focus everyone’s minds. The status quo is not going to last for ever if the port of Hodeida is not cleared.

“There is a narrative in Saudi, the United Arab Emirates and the Yemen government that you simply cannot trust the Houthis; they never do what they promise to do and that will be confirmed if this does not happen. But it requires flexibility from the government of Yemen as well, because they need to give the assurances that they are not going to walk straight in Hodeida as soon as the Houthis leave.”

Hunt travelled to Aden where he spoke to Yemen’s foreign minister, Khaled al-Yamani. He met Houthi negotiators including Mohammed Abdel-Salem in Oman at the start of his three-day visit to the Middle East.

Hunt said the Stockholm agreement made clear “that the independent police commissioner would control Hodeida and so it would not come under the control of the government of Yemen, so they were not handing over Hodeida, but they are nervous that the moment they leave Hodeida, the government will move in. They are obviously not prepared to accept that so it is a question of building up trust so that they do leave Hodeida.

“It was a bargain, a deal in Stockholm that the coalition will stop the aerial bombardment in return for the Houthis leaving Hodeida and that is the bit that has not happened. It is now 80 days so there is a real sense of urgency.”

He said the agreement was the last chance to achieve peace, and that he sensed neither side truly wanted to revert to an all-out war.

Hunt said he had a “very extensive” discussion with the new Saudi foreign minister, Ibrahim al-Assaf, “about human rights issues including JamalKhashoggi, the women’s rights issue and the jailed blogger Raif Badawi”. Badawi was sentenced to 10 years in prison and 1,000 lashes in 2014 for “insulting Islam through electronic channels”.

Hunt said he told Riyadh it needed “to make progress in all these areas in order to win the argument with the British public that reform is getting back on track”.

Asked if he thought there was a credible legal process inside Saudi Arabia to try those responsible for Khashoggi’s murder, Hunt said: “The jury is out on that and we have to see what happens, but what everyone needs to be confident of is that this is something that cannot and will not happen again.”