March's tax hike on Argentina's soy and grain exports sparked widespread protests and blockades by farmers and sporadic food shortages in the nation's capital, Buenos Aires.
So far, Julio Cobos, Kirchner's vice-president, who settled what had been a tied senate vote by voting against the bill, has said he will not resign, however there is growing pressure from within the governing coalition to do so.
Al Jazeera's Teresa Bo in Buenos Aires says there had been strong rumours of major changes to the government after the tax defeat.
Fernandez, who is close to Kircher and her husband, former Argentina president Nestor Kirchner, had reportedly tried to resign earlier, but had been prevented by the president in order to prevent the government from seeming weak, our correspondent says.
And it is thought the two resignations will not be the last in a long list of possible replacements, our correspondent adds.
Argentina is one of the world's biggest agriculture producers and half of its exports come from farms.
However Argentina is split over how to best distribute profits gained from the high prices on exports such as soy.
The bill had passed the lower chamber of deputies on July 5, despite facing stronger than expected opposition.
It proposed to raise grain export tariffs from 35 to 47 per cent, on a sliding scale linked to global prices.