Lendlease and Seven Trent Water are two Murphy clients that have assured the union that they will engage with Murphy about respecting workers鈥 rights.
Unite is pressing others to follow suit but not all are proving amenable.
The beef follows Murphy鈥檚 sacking of four Unite members in the Republic of Ireland last year.
The four members, one of whom was a shop steward, were fired 鈥 Unite says 鈥 because of their participation in a workplace meeting after Murphy International Limited (MIL), Murphy Group鈥檚 Irish subsidiary, violated national agreements on pay.
The only workers to be sanctioned by MIL following this meeting were members of Unite, while non-union workers who participated faced no penalties, Unite says.
Some of Murphy鈥檚 biggest customers, including National Grid and Network Rail, have rebuffed Unite and so far refused to challenge Murphy.
Unite general secretary Sharon Graham said: 鈥淚n the wake of the blacklisting scandal, we cannot allow union-busting employers like Murphy Group to go unchallenged. Murphy鈥檚 customers and potential future customers must offer more than empty pledges of support for workers鈥 rights 鈥 they must use their influence and demand the Murphy Four are reinstated.鈥
A Murphy spokesperson said: 鈥淛 Murphy & Sons acted in line with its legal requirements in response to unauthorised strike action at its Aughinish site in Limerick, Ireland.聽 The company has always indicated that it is open to resolving this matter amicably and sensibly.聽 We remain committed to maintaining an open dialogue with all parties around this issue to try to draw a line under this matter.鈥
Got a story? Email news@theconstructionindex.co.uk