Mr Robinson accused Mr McGuinness of a "nasty attack"
First minister Peter Robinson has said he regrets "that the deputy first minister continues to publicly make intemperate personal remarks about me".
He was speaking after Martin McGuinness suggested the first minister might be taking "cold feet" on a deal over devolution of policing and justice.
Martin McGuinness expressed anger at Mr Robinson's failure to meet Gordon Brown this week to discuss the matter.
Mr Robinson cited uncertainty over the future budget for devolving the powers.
He said there was no point in having a meeting while differences between budget estimates of the Treasury and local criminal justice agencies remained.
Mr McGuinness rejected Mr Robinson's argument and said the DUP should explain why they were not prepared to have more meetings this week.
"The people want to see local politicians provide for them through local accountability a first class policing service," he said.
He also said: "I have to record my annoyance at the fact I have not been able to develop a close working relationship with Peter Robinson.
"That is through no deficiency or lack of effort on my part."
In a statement on Wednesday evening, Mr Robinson said: "I am surprised that he believes such attacks are likely to help us resolve the issues that we presently face.
"I am not going to play his game by responding in kind.
He said Mr McGuinness was guilty of a "one-sided nasty attack".
Alliance leader David Ford said the pair needed to maintain a working relationship.
"People don't want more rows at the heart of government. They want to see an Executive that delivers."
Meanwhile, asked earlier if he would stand again for Westminster, Mr Robinson said the DUP were determined to phase out dual mandates.
However, he said his own position would be discussed with party officers.
Mr Robinson said some in his party believed the leader of unionism should be represented both at Stormont and Westminster during the fledgling years of the Assembly.
Also on Wednesday, Nigel Dodds, the DUP deputy leader, said the £8m cost of the latest Assembly expenses reinforced the DUP's argument for reducing the number of MLAs and streamlining the Stormont bureaucracy.