The centre, set up after the 1980 St Pauls riots, is based in a building owned by Bristol City Council and gets Â£37,500 funding a year.
The council said it was "committed" to the centre but was looking for a new organisation to run it from March.
Its manager said problems were being addressed and asked for more time.
The Malcolm X Community Centre Ltd (MXCC) was warned by the council in August of tenancy breaches including failings in financial management, noise and rubbish issues and subletting without consent.
On Tuesday, the council said it was seeking "expressions of interest" from groups wanting to take over the lease short-term. It also wants a group to "secure a lease and grant funding for the site from April 2017".
A spokesman said that MXCC would be supported to remain in the building and so "that they are able to submit an expression of interest for the lease" in future.
Manager Steve Stephenson told BBC Radio Bristol they had no full-time staff and needed more support, not new management.
"The Malcolm X Centre has had problems in the past but we have worked really hard for the last six to eight months to address many of these problems," he said.
"Give us an extension for another six months or so, so we can complete the task."
It follows an announcement that funding was being withdrawn from another local group, the St Pauls Afrikan Caribbean Carnival, after the council and Arts Council England said they lacked confidence in them.
Mr Stephenson said there was a "perception in the community that race plays a part" in the decisions but added: "I would say that we need to get our house in order - there are things we need to do that don't have anything to do with race."
The group running Bristol's Malcolm X community centre has been told its tenancy will be terminated after being warned of "substantial" lease breaches.