The government estimates there are four million stray dogs and the law, which will be debated over several days, will allow the killing of sick animals and those with "negative behaviour".
The maximum fine for abandoning dogs would be increased 30-fold to 60,000 lira ($1,800).
President Recep Tayyip Erdogan said ahead of the debate that Turkey faces a problem "like no other civilised country" which was "growing exponentially". A growing number of rabies cases particularly worries the government.
Authorities have denied however that they want a mass euthanasia. Erdogan said that people want "safe streets".
Animal rights groups have called for a mass sterilisation campaign and opposition parties have vowed to fight the law, even if it is passed in its current state.
The Republican People's Party, which controls Istanbul and other major cities, has said its mayors would not apply the law. Demonstrations have been held in recent weeks, including inside parliament.
The government has said mayors that refuse to carry out the law could be jailed. And it has banned visitors' access to parliament to avoid protests.
The debate has revived discussion of a 1910 measure under the Ottoman authorities when tens of thousands of stray dogs were rounded up in Istanbul and sent to a deserted island in the Marmara sea.
The dogs ate each other and most died of hunger.
