Mr Dalli, 64, became the EU's commissioner for health and consumer policy in 2010.
The controversial tobacco directive which brought down Mr Dalli will be revised, Mr Borg said.
Mr Dalli insists that he was unaware of any such attempt by the Maltese entrepreneur to buy influence.
He called Mr Dalli's "various complaints and accusations of illegal or incorrect conduct" by the commission and Olaf "incomprehensible".
His Maltese predecessor John Dalli left the Commission after being linked to a cash-for-influence scandal involving new tobacco legislation.
Meanwhile Health Commissioner John Dalli said that it would give the EU the highest standards of animal welfare in the world.
Former EU health commissioner John Dalli has insisted on his innocence in an alleged attempt to peddle influence, days after he resigned.
Mr Dalli has been preparing tough new EU legislation - a reform of the Tobacco Products Directive - to make smoking less attractive.
Commission Vice-President Maros Sefcovic has taken over Mr Dalli's duties on an interim basis until a new commissioner of Maltese nationality is appointed.
"Olaf found that Commissioner Dalli had taken no action to prevent or dissociate himself from the facts or to report the circumstances, " it said.
The EU Health Commissioner, John Dalli, took a swipe at the Germans over the release of inaccurate information which he said spread "unjustifiable fears".
Olaf said it had not found "conclusive evidence of the direct participation of Mr Dalli but did consider that he was aware of these events".
British Labour MEP Linda McAvan was more critical, and disagreed with Commissioner Dalli's claim that ethical considerations were outside of the scope of EU legislation.
Mr Dalli addressed MEPs following reports over the summer that meat from a cloned animal on a farm in the Scottish Highlands had entered the food chain.
In a statement on 19 October Olaf said its investigation had "found no conclusive evidence of the direct participation of Commissioner Dalli in the operation for requesting money".
He said Mr Dalli must respect Article 245 of the EU Treaty, which rules that officials can be stripped of their pension or other benefits if they fail to act with integrity.
Mr Dalli, a former Maltese cabinet minister, has argued that he is the victim of a tobacco lobbying campaign to block tough new legislation - the Tobacco Directive - to make smoking less attractive.
On Tuesday, the European Commission announced Mr Dalli's resignation, saying that the EU's anti-fraud office (Olaf) had established that a Maltese businessman had tried to use his contacts with Mr Dalli for financial gain.
But it went on to say that "there are a number of unambiguous circumstantial pieces of evidence gathered in the course of the investigation, indicating that Commissioner Dalli was aware of the activities of the Maltese entrepreneur and of the fact that this person was using the commissioner's name and position to gain financial advantages".
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