Relations between the two countries were further strained this week when Ukraine's parliament passed a law making the use of the Ukrainian language mandatory for public sector workers.
Russia says the move discriminates against Russian speakers in Ukraine – for many, particularly in eastern regions, Russian is still the first language.
The new tensions add to the challenges facing Ukraine's President-elect, Volodymyr Zelensky, who ousted Petro Poroshenko at last Sunday's election by a landslide.
What did Putin say exactly?
On Wednesday, the Russian leader announced the passport scheme would be applied to Donetsk and Luhansk, the self-declared republics seized by Russian-backed separatists in 2014 after Russia annexed Crimea from Ukraine.
Crimea itself was incorporated into Russia, giving its inhabitants the same citizenship rights as those in Russia.
This week he signed a decree that offers people in eastern Ukraine's separatist territories Russian passports in less than three months.
Now he says he may extend the scheme to the whole of Ukraine.
Ukrainian politicians accuse Russia of trying to make Ukraine's territorial divide permanent.
Mr Putin said people living in Donetsk and Luhansk who considered themselves Russian were entitled to Russian passports.
"For many years already – about 10 years – Poland has been giving out passports, Hungary has also been doing so, to Hungarians, and Romania… so are Russians living in Ukraine worse than Poles, Hungarians?" he said.
Mr Putin said earlier that people living in the rebel territories were "completely deprived of civil rights" and could not "move normally" or "realise their most elementary needs".
On Saturday, he announced: "We are considering providing a simplified procedure [of obtaining Russian citizenship] to all the residents of Ukraine."
What is Ukraine's position on the citizenship move?
In a Facebook post after Wednesday's announcement abut the Russian passport move, Mr Zelensky's team called Russia "an aggressor state which wages war against Ukraine".
"This [Mr Putin's] decree is not bringing us closer to achieving the ultimate goal: a ceasefire," it said.
The armed conflict in Ukraine has claimed about 13,000 lives since 2014.
In the run-up to his election, he had said he wanted to "renew relations" with eastern Ukraine and start a "powerful information war to end the conflict".
In response, Russia said it wanted him to show "sound judgement", "honesty" and "pragmatism" so that relations could improve.
What is the new language law?
Under the new law:
- Ukrainian must be used for signs, letters and in adverts
- Local TV channels are set a target of 90% Ukrainian content
- Ukrainian should be used for all official duties of public servants ranging from the president to judges, doctors and bank workers
Mr Zelensky, whose own first language is Russian, defended the new legislation, which his opponent Petro Poroshenko had championed.
However, he added that he intended to review it once he took office. He said he would conduct a "detailed analysis of this law to make sure that all constitutional rights and the interests of all Ukrainian citizens are respected in it".
