A Department for Business spokesman described the current parental leave system "old-fashioned and too rigid" after a study found few fathers are taking additional paternity leave.
The spokesman said: "This is why we are introducing a system of shared parental leave from April 2015 so that fathers can take more leave if they want to in the early days of a child's life.
"We want to challenge the myth that it is the mother's role to stay at home and care for children".
TUC General Secretary Frances O'Grady said a "good gift" this Father's Day would be for ministers to increase statutory paternity pay rates and for employers to top it up for longer.
Ms O'Grady said: "Poor levels of financial support are preventing new dads from taking extra time off and are particularly affecting low-paid fathers who simply cannot afford to take leave.
"Extending paternity pay from two to six weeks and paying a better statutory rate would make a massive difference, as has been shown in other countries".
He comments follow a TUC study that found few fathers are taking advantage of additional paternity leave, mainly because of the low statutory rate of pay they would receive.