The next PM has to be trusted to see Brexit through "promptly and sensibly", Foreign Secretary Jeremy Hunt has said.
Visiting the West Midlands, the Tory leadership contender said jobs depended on the right outcome and he was the "right person" to deliver it.
Tory members will decide over the next month whether Mr Hunt or Boris Johnson becomes party leader and PM.
Mr Johnson has said it is "eminently feasible" that Brussels could agree a new deal before 31 October.
The UK is due to leave the EU on that date, with or without Parliament backing the existing deal reached by outgoing Prime Minister Theresa May and other European leaders.
The EU granted the UK a seven-month extension in late March in the hope the parliamentary deadlock - which has seen MPs reject the terms of withdrawal three times - could be broken.
Mr Johnson is the favourite to succeed Mrs May after winning the support of more than 50% of his colleagues this week as MPs whittled the candidates down to the final two.
The party's 160,000 or so members will decide the next Tory leader by postal vote - with the result to be announced in the week starting 22 July.
Speaking on a visit to a factory in Worcester, Mr Hunt argued the contest was a test of character over Brexit, saying: "Do you have the skills and can you be trusted as prime minister to get the right outcome?
"Thousands of jobs in the West Midlands depends on having a wise prime minister making sensible calls as to how we leave the EU promptly, but also in a way that does not harm business. I am that person."
Both Mr Johnson and Mr Hunt have said Mrs May's withdrawal agreement is effectively dead and suggested they could get an improved deal from the EU.
Brexit Secretary Stephen Barclay, a supporter of Mr Johnson's, told the BBC this would prove a "significant challenge", given that the EU has ruled this out and leaders are not scheduled to meet again until 17 October.
"The big challenge is that you have got two hurdles," he told
"One is are there any concessions that will get the deal through Parliament, because Parliament has become increasingly polarised.
"And second, there is a timing issue. You have got to get the legislation through the House of Commons. And if you look at precedent, at things like Maastricht, then that will be a significant challenge just on the timescales from the 17 October EU Council."
Speaking in Brussels, European Council President Donald Tusk reiterated that the terms of the UK's withdrawal agreed with Theresa May last autumn were not up for renegotiation.
While the EU did not want the UK to leave without a legal agreement, he said the bloc would only be open to further talks on its future relations if the UK's position "evolved".