The demand comes from Don Foster, Liberal Democrat MP for Bath and the co-chair of the party's backbench committee for Culture, Media and Sport.
In a letter to Ecclestone, dated August 31 and posted on Friday on his website, bathlibdems.org.uk, Foster believes the new BBC/Sky deal does not "promote the best interest of license-fee payers and F1 fans".
At the end of July, Ecclestone announced that from next year Sky Sports will screen all practice, qualifying and grands prix live.
The deal between the two broadcasting giants, over seven years and worth an estimated £45million per annum, will see the BBC show just 10 races live, with the remainder likely to be in aired full at a deferred time.
The decision prompted an uproar from F1 fans in the UK, furious at the fact they will now have to pay to the subscription service if they are to view all 20 races live next season.
Foster has expressed particular concern over how the deal transpired, citing the "apparently divergent views of FOM (Ecclestone's Formula One Management) and the BBC about the sequence of events".
Foster, who has also written to BBC director general Mark Thompson, quotes Ecclestone from one source stating he wanted F1 "to stay free to viewers".
Foster further quotes the 80-year-old as saying: "If they (Channel 4) had said they wanted to sign a contract to start broadcasting for £45m a year, then we would have probably done it."
In particular, the discrepancies over whom instigated the deal angers Foster, with Ecclestone apparently claiming "the BBC brought Sky to us with the idea of a joint contract. It was not us who made that decision".
In contrast, Foster highlights Neil Land, BBC Sport chief advisor and business manager as saying: "FOM decided a broadcast partnership between the BBC and Sky was in the best interests of the sport."
In his letter, Foster adds: "I am alarmed that there is such a gulf between your stories.
"This deal has led to disappointment and anger among F1 fans and now they have to sift through completely contradictory accounts of who decided what.
"The least fans deserve is a clear explanation of what has happened. I urge you to give it."
Speaking to Press Association Sport in outlining his motivation behind the letters, Foster said: "We didn't feel like the views of the fans were being represented, so we started to look into it and found two completely contrasting stories.
"A lot of people are angry about this. When the public isn't getting the clear explanation they deserve, we need to apply pressure.
"There are serious questions to be answered, which I have sent to the relevant parties, and we will now see what comes back.
"We're giving them the opportunity to explain themselves."
It is understood the Commons Select Committee for Culture, Media and Sport are to debate the issue at a meeting next month when senior BBC figures are due to be questioned.
"That is a logical step and if they feel it is an issue worth pursuing then I would welcome it," said Foster.
"As Liberal Democrats we consider ourselves strong supporters of the BBC. It is an important organisation, one of the best broadcasters in the world.
"It is because we feel it holds such an important position that it really needs to be able to withstand this kind of scrutiny.
"If fans, who are also licence fee payers, are feeling so put out by this, then questions need to be asked of the BBC as to whether their conduct contributed to that.
"Was there a better outcome they could have contributed to than the one we have which has left so many people unhappy?"