According to the Welsh secretary, standards in public life were "a real problem" under Boris Johnson's leadership.
Robert Buckland stated that he would like to see a "quick return" to better standards under the next prime Minister.
Stephen Crabb MP, an ex-government minister, said that he had always "real questions" about Johnson's leadership abilities.
Plaid Cymru and Labour called for a snap election.
Boris Johnson was resigned earlier this week after he called for the resignation of the Llanelli-born South Swindon MP. He filled the vacancy as Welsh Secretary to "serve the nation" and not to "prop Boris Johnson up."
When asked by BBC Politics Wales whether standards were lower under Boris Johnson's leadership he replied that standards were a "real concern and matter of debate".
He said, "I believe the Conservatives should embody high standards in public life."
He said, "I believe part of the tradition that is Conservative is about respecting institutions, respect for office and respect for political enemies, and I want to see an immediate return to that."
Buckland stated that he would like to have a business-like relationship "with the first minister of Wales and the government" and to "not only fly the flag for Wales, but to attract as much investment as possible.
As there are more leadership candidates, Mr Buckland didn't rule himself out of running for the position.
He said, "Well, who knows? - I've learned to expect the unexpected."
Stephen Crabb was the Welsh Secretary and Work and Pensions Secretary during David Cameron's tenure. He said that he was not a big believer in Boris Johnson.
Talking to Politics Wales, Mr Crabb stated that he didn't vote for him in the 2019 election as he was unable to manage an efficient and effective administration.
"He clearly won an amazing mandate in the 2019 general elections... which was impressive, and was one thing that did bind them together.
He said, "But what has been happening over the past eight months in particular is a steady draining away trust and goodwill."
Boris Johnson's "very puny, very pugnacious" style was not for him.
He said that he wasn't sure if he could simplify very complex, difficult problems into simple slogans or propositions for policy direction.
The Preseli Pembrokeshire MP stated: "I don’t want any Conservative candidate going out and trying to pretend really big, hard challenges are easy and can easily be solved with just a few words and a couple of simple slogans.
He said, "We need to be honest about the nature and challenges facing the country."
"I want to hear their plans for the union. I believe there has been insensitivity over different parts of the union at times. I want to see a new Conservative leader who can be a persuasive face for the UK government in Scotland and Northern Ireland."
While Mr Crabb is still deciding which candidate to support in the race, Montgomeryshire MP Craig Williams supports ex-Chancellor Rishi Sonak.
Williams, who was a UK government aide for the Treasury, resigned earlier in the week. He said: "I've seen firsthand [Mr Sunak]'s work ethic and I believe a teetotal workeraholic is something that the country might require right now.
Jo Stevens MP, Labour's Shadow Welsh Secretary, said that "Why should the people put up with [Boris Johnson]for another three months, squatting down in Downing Street when he's not fit to be prime minister?"
"What damage, or more damage is he going do in the next three-months?"
She said, "He must go now, and we need not only a change in the top of government; we need to create a new government. That's why we need general elections."
Plaid Cymru's Westminster leader Liz Saville-Roberts MP stated: "Surely, if Conservative MPs see the light with Boris Johnson, and they want him to go, he should be gone now in the here and then to ensure that there's no malign influence with next prime minister.
She said, "We urgently need, especially in Wales, where we've been the brunt of Tory cuts. We really need a change of government now, because Tory government have run short of ideas and have done enough damage over the past 12 years or so."
