The pair had opened talks before Christmas after Medco suggested it would consider tabling an offer nearer to £430mln, before reducing this to £401mln shortly after.
Ophir said the board had met to discuss the offer, which was made on Friday and had “unanimously agreed” to turn it down, without giving any further details.
Under City takeover rules, Medco now has until 28 January to either make a firm offer or walk away from a deal altogether.
Ophir, which is due to publish a trading update on Tuesday, lost more than half of its value last year after failing to find financing for a liquefied natural gas (LNG) project in Equatorial Guinea.
The company is expected to write down the value of that project by US$300mln and also move its headquarters from London to Asia, where it bought a package of oilfields last year which doubled its annual production.
Ophir shares were down 2.6% to 43.8p in early deals on Monday.