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".
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.