Melbourne's lord mayor wants to introduce an extreme weather policy for the vulnerable, after a string of concerning reports and incidents involving the city's homeless.

Robert Doyle said the stabbing death of Wayne ''Mousey'' Perry and January's protracted heatwave had been ''wake-up calls'' about the hardships faced by Melbourne's rough sleepers.

He said he was heartened by the concern shown by Melburnians following a Fairfax Media report about homeless people being chased from airconditioned public spaces.

''It still makes news in this city, [while] in other cities this passes as unremarkable,'' Cr Doyle said.