Industry speculation suggests that veteran chairman Gianluigi Aponte might be about to step down from the world's second-biggest ocean cargo mover and that the company is said to be preparing an announcement within days.
Rumours that Mr Toft had left Maersk to join MSC started almost immediately, but MSC has declined to comment, reports Lloyd's List of London, citing the Wall Street Journal quoting unidentified people with direct knowledge of the matter.
Mr Aponte is said to be stepping down as executive chairman of MSC to be replaced by his son, Diego, according to the Wall Street Journal.
Maersk announced last week that Mr Toft was leaving his post with immediate effect to pursue opportunities outside the company.
Not since Flemming Jacobs quit Maersk 20 years ago to join Singapore's Neptune Orient Lines has the departure of a top executive from the Danish group created such a stir.
Adding spice to the news was speculation that Mr Toft, who had been seen by some as a successor to AP Moller-Maersk chief executive Soren Skou, would be joining MSC. The rumours started almost immediately, but family-owned MSC, where Diego Aponte is group president and chief executive of the container shipping division, has declined to comment.
But that has not stopped speculation about why Mr Toft would leave the world's largest containership operator to join its biggest competitor, albeit also a partner through the 2M alliance.
Yet he would have built up a close relationship with MSC through that partnership, and also on work to make containerships safer by cracking down on cargo misdeclarations. This is an issue about which both companies feel very strongly, and on which Diego Aponte and Soren Toft have been collaborating to find a solution through an initiative sponsored by the Box Club.
One line of thought is that Diego Aponte could move up to executive chairman, with his father, now in his late 70s, easing up. Having said that, those who know the older Mr Aponte think it is highly unlikely that he will take much of a step back.
Suggestions that Diego Aponte could take over the cruise division that is currently run by his brother-in-law Pierfrancesco Vago are considered less likely by those who know the family.
For Mr Toft, the attraction of becoming chief executive of MSC's containership operations could be far more freedom in a privately-owned group without the limitations of outside shareholders and stock market demands.