Send a running function to background in bash

As the title says, I want to send a running function to the background. Something like

function myfunc {
    some_command | while read line
    #some_command must be running indefinitely
    do
        #if some condition on ${line} satisfies
        #Do something and go to background
    done
}

Is it possible?

I know that it is possible to call a function directly & to directly run it in the background. That is pretty obvious. This is not duplicate of this.