The centre-left leader proposes reforms to tackle corruption and create jobs, possibly with the backing of comedian Beppe Grillo, whose Five Star Movement won an unexpected 25% in the election and holds the balance of power.
Marc Bayliss, the council's deputy leader, said the authority just wanted to "create balanced and fair communities with a mix of all sorts of residents".
Welsh Liberal Democrat leader Kirsty Williams said Wales needed "a proper functioning parliament with additional responsibilities to create a stronger economy in a fairer society".