The North African Muslim country, which toppled autocrat Zine El Abidine Ben Ali in 2011, grants women more rights than other countries in the region, and since last year has allowed Muslim women to marry non-Muslim men.
But in a show how divided society remains, thousands demonstrated on Saturday in front of parliament against any changes to inheritance rules.
The current system is based on Islamic law which typically allows men to inherit double what a woman would receive.
“I propose equality inheritance to become law,” President Beji Caid Essebsi said in a speech.
But in the face of the opposition from conservatives, he left the door open for some exceptions, saying families who wished to continue the allocation based on Islamic law would be able to do so.
Parliament now needs to decide on a bill.
Tunisia is ruled by a coalition of moderate Islamists and secular forces which have been managing its democratic transition since 2011, avoiding the upheaval seen in Egypt, Libya or Syria.
They had agreed in 2014 on a constitution granting far-reaching political rights, limiting the role of religion and holding free elections, which stands out in a region often run by autocrats.
But one of the few areas where the Islamists have resisted change is the inheritance law.
To break the standoff Essebsi, a secular politician, had in August 2017 set up a committee to draft proposals to advance women’s rights, winning praise from secular-minded women.
While Tunisia has been hailed as the only “Arab spring” success story economic growth has been disappointing, however, with high unemployment driving many young Tunisians abroad who had joined the uprising.