Britain is the EU's biggest financial center and used by companies from across the region to raise funds.
The UK is due to leave the EU in March, triggering renewed efforts to push through reforms to create a deeper EU capital market by strengthening pan-EU supervision.
Some continental European political parties have traditionally been suspicious of "Anglo-American" capital markets and fear that Britain will row back on financial rules after Brexit to keep London competitive as a global financial center, a charge the UK government has rejected.
Stronger EU watchdogs would also play a more central role in ensuring that foreign firms continue to abide by rules that are "equivalent" or as strict as those in the bloc once initial access has been granted.
"Especially in the context of Brexit, progress on these reforms and an enhancement of the equivalence regime are urgently needed to pave the way for a single supervisor for the capital markets union," said Pervenche Beres, a French center-left member of European Parliament.
But the draft law also needs approval from EU states or "council," and some are wary of giving more power to EU bodies.
"It's a very sensitive subject. There are national interests and European necessities at stake here," Karas said.
"It's a good starting point for what are going to be very difficult negotiations with the Council as the Council and parliament's positions seem to be rather distant at the moment."
Lawmakers want a final deal by May to avoid a delay due to European Parliament elections and the appointment of a new European Commission.
If passed, the European Securities and Markets Authority would be able to intervene if it thought that too much business was being "delegated" or outsourced from the EU to firms outside the bloc.
This has emerged as a key issue in asset management given that money managers in Britain run funds holding trillions of euros across the EU.
The rules would also give more power to EU watchdogs to stop the sale of harmful financial products and scrutinize the conduct of people who work in finance more closely.
Ahead of Brexit, the EU has also begun tightening market access conditions for foreign clearing houses and investment banks, many of which are in London.
(Reporting by Huw Jones. Editing by Jane Merriman)