It was expelled from the United Nations council last year after its forces invaded Ukraine.
Diplomats said Russia was hoping to regain some international credibility after being accused of human rights abuses in Ukraine and within its own borders.
The latest evidence of those abuses was presented to the human rights council on Monday in a report from its Commission of Inquiry on Ukraine. Erik Mose, chair of the commission, said there was continuing evidence of war crimes including torture, rape and attacks on civilians.
A separate report two weeks ago by the UN's special rapporteur for Russia, Mariana Katzarova said the human rights situation in Russia had also "significantly deteriorated", with critics of the invasion subjected to arbitrary arrest, torture and ill treatment.
Russia is seeking to rejoin the United Nations human rights council in an election that will be seen as a key test of its international standing.
In the document seen by the BBC, Russia promises to find "adequate solutions for human rights issues" and seeks to stop the council becoming an "instrument which serves political wills of one group of countries", understood to be a reference to the West.
In the next elections, due on 10 October, Russia will compete with Albania and Bulgaria for the two seats on the council reserved for central and eastern European countries.
The Russian position paper - circulated at the UN - says it wants to "promote principles of cooperation and strengthening of constructive mutually respectful dialogue in the council in order to find adequate solutions for human rights issues".
Its core pitch is that Russia would use its membership "to prevent the increasing trend of turning the HRC into an instrument which serves the political will of one group of countries".
The shadow foreign secretary, David Lammy, said Russia had committed atrocities in Ukraine, its leader had been indicted for war crimes by the International Criminal Court and had shown utter contempt for the UN Charter.
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Navalny, who is serving two sentences on politically motivated charges of embezzlement and extremism in a penal colony in Melekhovo in central Russia’s Vladimir region, said that he had been transferred to what he described as a “single cell-type room” for 12 months due to his “incorrigibility”.
Moskalyov has reportedly been given two weeks in an isolation cell, and told his lawyer that he is feeling well but that he often gets tired as he’s only allowed to use his bed between 10 PM and 6 AM.
“This time around, he was sent to solitary confinement for resting his head on his hand while sitting at a table. He was accused of ‘turning the table into a bed’,” Russian human rights organisation OVD-Info noted on Wednesday.