Mascherano and Begovic have found themselves out of their respective sides in recent days amid talk of being unsettled by interest from other clubs.
Former England boss Taylor feels the clubs must act - and that the Professional Footballers' Association must back them.
"Both players should - and must - be punished for their behaviour, and the players' union, the PFA, should support whatever action the clubs wish to take," he said in his Daily Express column.
"The standing and reputation of professional football and footballers are constantly taking a public battering.
"A lot is undeserved, but effectively refusing to play for your club is tantamount to treason - and the punishment should fit the crime."
He continued: "Both Mascherano and Begovic will have to do a lot of explaining to supporters of their clubs to justify their actions, but do they care? Not one jot.
"In their minds it is all about their own interests. That they are contracted to their clubs to play for them does not enter their thinking.
"Their actions are a disgrace to their profession."
Source: Team Talk
Source: Team Talk