Heard is facing claims that she illegally imported her two dogs into Australia.
One website article made inaccurate references to how much the case has cost her husband.
Defamation proceedings were drafted and set to be issued on the couple’s behalf by Belfast-based lawyer Paul Tweed.
However, Mr Tweed, of Johnsons Solicitors, confirmed on Wednesday that two cases have now been “settled amicably”.
The outcomes involve an apology being issued by the Belfast Telegraph about unfounded allegations relating to “grossly exaggerated expenditure”, he said.