"I have no hesitation in accepting my relationship with Amrita Rai. She and her husband have already filed a mutual consent divorce case...Once that is decided we would formalise it," the former Chief Minister of Madhya Pradesh tweeted.
"But I do condemn encroachment in our private life," the Congress General Secretary added after media reports about their relationship appeared.
Singh's wife Asha passed away last year after prolonged illness. He has four daughters and a son from his marriage.
Rai, said to be in her early 40s, also tweeted that she will marry Singh after divorce with her husband.
"I have separated from my husband and we have filed a mutual consent divorce papers. After which I have decided to marry with Digvijaya Singh," she tweeted.
Rai alleged that her email account and computer were hacked.
"My email/computer (have) been hacked & contents tampered with. It is a serious crime in India & encroachment of my privacy. I strongly condemn it," she said in her tweet.
Congress General Secretary Shakeel was asked about the issue at the party's media briefing but he said he was not aware of it.
BJP, on the other hand, raised the question of morality and legality and asked Congress leadership to take cognizance.
Amrita Rai's husband said the relationship between the couple had ended a long time back and that she was free to take her own independent decisions.
"We have been separated for some time now and applied for the mutual consent divorce. Relationship between us has ended long back and Amrita is free to take any decision regarding her life and I will respect it," he posted on the facebook.
Rai's husband also appealed everyone to respect their privacy and not to create an embarrassing situation regarding the issue.
He, however, said it was a "tough and painful phase" for him.
"Since it is a legal procedure, it takes its own time but our relationship has ended long time back. I wish her well for her future. I know that my friends, well wishers and students are sad due to this but I also know that they are standing by me," the husband said.
Attacking Digvijaya Singh, Lekhi said, "Secret marriage is not possible because there is no divorce. All I can say is that the lesson of morality gives a perverted definition and I think Congress leadership should take into cognisance because I cannot react more than this as I am not Mr Digvijay Singh."
Lekhi, an advocate, also raised the legal issues.
"By law, it is also an offence. There can be punishment as per law, as there is no divorce yet. It is for the husband to act. There is a criminal activity involved and the husband has to act," she said.
She added, "A new definition of morality has been given by the actions of senior Congress leader Digvijay Singh Sahab. Those who preach lessons of morality should first look within themselves to see if they are practising what they are preaching to others. As far as the legal aspect of the matter of adultery is concerned, it is up to the husband of the lady to pursue the matter."
AAP leader Somnath Bharti said the "affair" is dangerous and adulterous.
"Its disgusting n dangerous @digvijaya_28 in affair with 43 yr old married woman and TV anchor @amrritarai, how mny such politicians r there
"Friends who are siding with @digvijaya_28 ji, plz understand that he has violated law as well of adultery and don't know when this affair started," he tweeted.
He also said that such relationships could be dangerous.
"There was one Pamella Bordes in UK who had got many politicians there trapped and country's many secret info was compromised. Plz think.
A politician when in power gets 2 much of authority in hand, lacs of crores worth of contracts get decided by him, secret info he possesses. As an ordinary citizen one hs more freedom than someone who gets into politics, recall what Bill Clinton had to go through for monica issue," the former law of minister of Delhi said in a series of tweets.
"A characterless politician is the most dangerous weapon in the hands of enemies to the nation. @arunjaitley ji had earlier stated that," he tweeted.