Earlier, the rupee opened 63.70 as against last closing level of 63.56 at the Interbank Foreign Exchange market and later dropped further four month lows to 63.77 before quoting at 63.63 at 1100hrs.
The rupee hovered in a range of 63.77 and 63.55 during the morning deals.
Sustained capital outflows bearing on the rupee sentiment, a dealer said.
Overseas, in New York the dollar started the week on the defensive in early trade after more disappointing US economic data reinforced expectations the US Federal Reserve will not hike interest rates any time soon, while concerns about Greece's ongoing debt talks pressured the euro.