Bridgewater Community Healthcare (BCH) said it had "made a lot of progress" since an inspection last summer.
But Labour MP Rosie Cooper said it was "staggering" that BCH was due to take over most of Liverpool's community health services from July.
The Department of Health is yet to respond.
The Care Quality Commission (CQC) conducted its first full inspection at BCH - which is used by about 1.5m people in north-west England annually - in May and June 2016.
It said it measured 40 domains across the services with one rated as outstanding, 27 as good and 12 as requiring improvement.
Overall, the trust received a rating of "requires improvement".
Ms Cooper, MP for West Lancashire, said she called on the CQC in July to publish its inspection report ahead of any decision on "awarding the multimillion-pound contract for Liverpool community health services".
She said: "What this report tells us is Bridgewater Community Healthcare needs to improve the services they have currently got."
Last November, BCH was chosen to run most of the city's community health services by NHS England and Liverpool Clinical Commissioning Group (CCG).
Ms Cooper said that the inspection rating "raises some very serious questions about the entire transaction process in Liverpool".
"I have called on the Secretary of State to review this sorry state of affairs and intervene to uphold NHS rules," she added.
Colin Scales, chief executive at BCH, said: "All the essential actions the CQC has asked us look at have already been addressed since the inspectors were on site, so we've made a lot of progress and are in a stronger position now as we move forward."
Katherine Sheerin, chief officer for NHS Liverpool CCG, said: "Bridgewater NHS Foundation Trust was identified as the preferred provider of community services in Liverpool because we believe it is the best organisation to help accelerate our Healthy Liverpool plans for making more care available in the community so that people do not end up in hospital."
She added the CCG was "confident the Trust is already taking action to address the issues which have been identified".

Michael Ellerton said steroid hormones prednisone and prednisolone appeared in his system because of mouth ulcer medication he was taking.
UK Anti-Doping (Ukad) refused his retrospective application for a therapeutic use exemption.
But Ukad added it was "satisfied Ellerton did not act intentionally".
Ellerton tested positive in an in-competition test at the Port Talbot Wheelers Cycling Club's annual 25 Mile Time Trial on 11 September 2016.
A ComRes poll for BBC Sport released earlier this week of more than 1,000 members of sports clubs and teams found more than a third (35%) of amateur sports people say they personally know someone who has doped.

The Labour-run council has issued £1.3m fines since August when traffic was banned from Coppergate and Lendal Bridge during the day.
Last week, a traffic adjudicator ruled the roads did not qualify because bus lanes and signage were inadequate.
But James Alexander, the leader of City of York Council, said "the principle of the trial" had been right.
Residents, tourists and businesses had complained about the restrictions from the outset.
Councillor Keith Aspden, Liberal Democrat leader, said: "It was a botched trial from the start which has made congestion worse and damaged local businesses."
In a statement on Monday night, Mr Alexander said lawyers had confirmed the implementation of the trial in August "was legal".
He denied its purpose had been to generate revenue, and said it had been aimed at reducing traffic over the bridge and through the city centre.
"It is clear, however, that the trial has been polarising and we need greater consensus amongst residents and businesses over measures to tackle congestion," Mr Alexander said.
He said he wanted an independently-chaired cross-party commission to look at the data and come up with more consensual suggestions for tackling congestion.
An MP has called on the government to review a decision allowing an NHS trust that "requires improvement" to run community health services in Liverpool.
[NEXT_CONCEPT]
A 55-year-old amateur cyclist has been banned from all sport for two years after testing positive for banned substances at a race in September.
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Traffic restrictions on York's Lendal Bridge will be lifted on Saturday, the council has confirmed.