The ruling Popular Party on Friday asked the Socialists for a meeting on the issue following mass shootings at the French satirical magazine Charlie Hebdo and bloody hostage drama at a Jewish supermarket in a related incident.
"We are going to propose that the government transpose the 'spirit of Paris' to the reforms regarding terrorism. Lets not divide here what united us in Paris," the spokesman for Socialists in the lower house of parliament, Antonio Hernando, told reporters.
He was referring to the historic outpouring of unity that saw nearly four million people rally across France on Sunday, some 1.5 million people in Paris alone.
Hernando will meet with Interior Minister Jorge Fernández Díaz and Justice Minister Rafael Catalá later on Tuesday.
"We should adapt the penal code to this new type of terrorist menace, which is very different from the conventional one," Díaz said during an interview with private radio Cadena Cope before the talks.
Under the changes suggested by the PP, the relevant section of the penal code will include mention of individual terrorists. To date, the law has focused on terrorist groups.
Both parties back the adoption of draft European legislation forcing airlines to hand over the data of passengers entering or leaving the European Union, known as the Passenger Name Record Directive.
But Hernando said the Socialists want to ensure that "all precautionary measures" are taken to ensure that individual freedoms are respected if the measure is adopted.
The Socialists have disagreed with a reform of the penal code presented by the Popular Party last month which included several controversial measures in addition to amendments aimed at strengthening the fight against terrorism.
To fight jihadism the Popular Party reform proposes imposing jail terms to those caught travelling to a territory "controlled by a terrorist group" if they have contacted and shown their desire to cooperate with that group.
The reform proposed by the Popular Party, which is still under discussion at a parliamentary commission, also calls for jail terms for anyone who trains others how to use weapons and explosives in connection with a terrorist group, even if it is only done online.
The intention to carry out an attack would not have to be proven for a penalty to be applied.
"We have a majority in parliament, that is enough to approve it, but we think it would be good if the Socialist Party was also in agreement," Díaz said.