Delyn MP Rob Roberts is facing allegations of inappropriate behaviour towards two junior members of parliamentary staff.
Messages seen by BBC Wales show the MP inviting a 21-year-old female intern to "fool around" with him.
He has also admitted asking out another House of Commons employee, which led them to move position.
The Conservative Party confirmed an investigation into his conduct was ongoing.
BBC Wales understands a complaint has been made about Mr Roberts's behaviour to Parliament's independent complaints and grievance scheme.
The MP said he recognised the request to the staff member "was inappropriate".
But he did not provide a comment about the messages sent to the female intern, when asked for a response.
Labour and Lib Dem politicians called for the allegations to be investigated and for the Conservative Party to remove his party whip.
Mr Roberts was elected as the Conservative MP for Delyn in north Wales in December 2019.
His seat was a gain from Labour in the so-called Red Wall and had been previously held by Labour MP David Hanson since 1992.
Mr Roberts propositioned the now former intern, who was working in Parliament at the time, in a series of messages in April this year.
In the messages, Mr Roberts referred to the intern's "picture on Twitter earlier. Lovely legs".
"Don't ignore me when I'm making you feel better," he told her.
The MP then went on to say that in future she might want to "fool around with no strings, you might come and visit me in London".
'I felt really vulnerable'
Later in the exchange the MP, who has recently split from his wife and come out as gay, told the female intern that he "might be gay but I enjoy… fun times".
The woman, who did not respond to the request, explained to the MP that she was in an "awful state" and was struggling with her mental health, to which Mr Roberts responded: "I was just thinking about fun times… Maybe if you thought of them too it might help you."
Speaking to BBC Wales, the former intern said the exchange made her feel "incredibly sick".
"I had had one of the worst days ever, and then I messaged him saying I'm crying, I'm not okay - and his solution to that was to proposition me," she said.
"I felt really vulnerable and I felt like I was being used to make him feel better about everything."
In a separate series of messages also seen by BBC Wales, Mr Roberts admitted asking another member of staff in Parliament out, resulting in them moving positions.
He said he told a male parliamentary staffer "that I liked him and asked him to dinner".
"Really poor judgement on my part, hands up. He didn't react very well because it created a problem for him that I felt that way," he said.
In a further message, Mr Roberts added: "The chief whip was involved, lots of drama."
What has the MP said?
In response, Rob Roberts told BBC Wales that earlier this year he invited a parliamentary staff member to dinner.
He said: "My invitation was not accepted and the staff member did not feel comfortable about it.
"I do now recognise that it was inappropriate to have extended the invitation to a member of staff in the first place.
"However, I have found the last few months and the 'coming out' process to have been particularly challenging and the cause of a great deal of mental stress to me.
"Creating a story from this interaction which has been dealt with amicably seems calculated to add to that stress, which is distressing."
What has the reaction been?
Welsh Labour Gower MP Tonia Antoniazzi tweeted: "The allegations about Tory MP Rob Roberts are very concerning and should be fully investigated."
She said it was "unclear why he still holds the Tory whip".
Jess Phillips, Yardley Labour MP, called for the party to deal with it as "swiftly as they dealt with Julian Lewis" - the Tory backbencher whose whip was removed after beating Chris Grayling to chair of the Intelligence and Security Committee.
Acting Liberal Democrat leader Ed Davey tweeted that the "allegations are incredibly serious. We must make sure women are safe at work".
"The Conservative Party must immediately remove the whip while they investigate, or they risk sending the message that this is acceptable conduct for a sitting MP."
What has the Conservative Party said?
Following the opposition calls, a Conservative Party spokesperson said: "An investigation into Rob Roberts MP's conduct is ongoing.
"While Mr Roberts has apologised for his behaviour and has been undertaking safeguarding and social media protection training, we are clear such conduct is completely unacceptable."
www.bbc.co.uk/news/uk-wales-politics-53472289
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