Doug Beattie, who was elected leader in May, was accused of racism and misogyny when old tweets, written before he entered politics, resurfaced.
They came to light after Mr Beattie, a former army captain, posted a separate joke tweet on Saturday in which he referenced the wife of a political rival and a brothel.
The historic tweets included derogatory comments about women, Muslims, members of the travelling community and people with mental health issues.
The Upper Bann MLA admitted that the posts were “pretty horrific” and that he was “deeply ashamed” of them. He denied being a racist or a misogynist.
“I will speak to my MLA group today and I will speak to my party officers through my chairman Danny Kennedy and if either group feels I should step down, then I will,” he told BBC Radio Ulster on Tuesday.
“Likewise, if they think I should refer myself to the party executive or the wider council on a vote of no confidence then I shall do that as well and the party will decide whether or not they can follow my leadership.”
Asked how close he came to resigning, Mr Beattie added: “On the cusp. I have a few very close friends who rang me – because I had withdrawn – and gave me advice.
“I offered my resignation and people said, ‘Let’s engage and see where we go’.”
Mr Beattie suggested the “dark and black humour” he used may have been a “coping” mechanism and reflective of him being “desensitised” by battlefield experiences.
“My mental health has been affected by what I have seen and what I have done,” he said.
Mr Beattie insisted he is not using that potential explanation as an “excuse” for his tweets.
“I don’t want anybody to think I’m giving an excuse because I am simply not, it was wrong and I am deeply sorry,” he said.
He said he was asking his party for a second chance.
“Whether or not my party feels that I am the leader who can lead them into the election and beyond will be their decision and I will abide by that decision,” he said.
“It the party want me to stay I will still be the leader next week, if they don’t want me to stay I will not be the party leader next week.”
He added: “I am asking people to look at the person I am now and not judge me on the person I was 10 years ago.”
The furore erupted after Mr Beattie posted a joke on Twitter on Saturday evening that referenced the wife of a political rival and a brothel.
DUP Stormont minister Edwin Poots said his wife was “disgusted” by the joke.
Mr Beattie apologised for the post and deleted it.
He reiterated that apology on the floor on the assembly on Monday, insisting he was “truly sorry”.
However, the controversy then escalated after the focus turned to Mr Beattie’s historical conduct on Twitter.
Mr Beattie was defended by two female politicians in his party, including assembly election candidate for North Down, Naomi McBurney, who said she respected Mr Beattie for what “he has delivered and can continue to deliver for Northern Ireland”
Several opposition parties criticised Mr Beattie over the tweets.
Naomi Long, leader of the Alliance Party and the Stormont justice minister, said she was “genuinely shocked at just how casually misogynistic” the tweet about Mr Poots’s wife was.
It would “have made many a 1970s comic blush”, she added.
The controversy has marked a dramatic turn of fortunes for Mr Beattie.
He posted the tweet on Saturday evening hours after a newspaper opinion poll indicated he was the most popular political leader in Northern Ireland.
He said his confidence had been rocked by the controversy and he had isolated himself from friends and loved ones.
“I have to pick myself up and come out of the shadows and face this head on,” he said.