Melenchon's emergence over the past week has raised the possibility that he will square off against far-right leader Marine Le Pen in the election's decisive second round in May, making the final result unpredictable.
"Two weeks ago, investors were starting to get comfortable with the idea of a Macron victory, but with the rise of Melenchon this is on the verge of becoming a four-horse race," said Rabobank strategist Lyn Graham-Taylor.
"A Melenchon win would be a very market-unfriendly event," Graham-Taylor added. "Enough reason to go short on France if you weren't already."
The gap between French and German bond yields has shot wider in recent months on the possibility that Le Pen will win the keys to the Elysee Palace and try to take France out of the euro single currency bloc.
The spread came off its February highs as centrist Emmanuel Macron made gains and polls suggested he would go face to face with Le Pen in the second round and win comfortably.
But recent polls have shown the race tightening as the front-runners faltered and Communist-backed maverick Melenchon surged after strong performances in two televised candidates' debates.
France's 10-year bond yield rose as much as 7 basis points to 0.95 percent, stretching the gap with German equivalents to 72 basis points, its widest since Feb. 27.
Investors also dumped low-rated South European government bonds such as Italy's on Monday with yields on 10-year bonds up 2 bps on the day.
These "peripheral" bonds tend to perform badly when there are concerns over the future of the euro zone.
A heavy week of supply could also be weighing on government bonds in a shortened week when liquidity would naturally be low anyway, said Lenk. Five euro zone countries are due to sell bonds via auctions this week.
There were yield gains further up the ratings curve as well, but monetary policy expectations and increased geopolitical risk pushed German 10-year yields a touch lower to a fresh five-week low of 0.213 percent.
Commerzbank analysts attributed this to reduced expectations for rate rises, prompted by dovish comments from European Central Bank policymakers.
ECB President Mario Draghi said on Thursday he saw no need to deviate from the ECB's stated policy path, which includes bond-buying at least until the end of the year and record-low rates until well after that.
Demand for low-risk assets also rose because of US missile strikes on Syria which have increased tensions in the Middle East and worsened Russia-US relations.