The United States decided last week not to renew exemptions from sanctions against Iran granted last year to buyers of Iranian oil, taking a tougher line than expected. Oil prices rose on concerns of a tighter oil market.
Falih was speaking to Russia's RIA news agency on Tuesday, without specifying whether, or by how much, output levels could change after June.
The Saudi minister's comments came after US President Donald Trump said last week he had called OPEC and told the group to lower oil prices, without specifying to whom he spoke or whether he was referring to previous discussions with OPEC officials.
Oil prices have surged by almost 40 percent since January, lifted by the OPEC+ supply cuts as well as by US sanctions on producers Iran and Venezuela. Brent crude futures were trading at $72.40 at 0855 GMT on Tuesday.
Falih, commenting on Trump's statement, told RIA that the world's top oil exporter was ready to meet consumer demand after the Iran oil waivers expire in early May, including by replacing Iranian oil with Saudi supplies.
But Riyadh will not voluntarily exceed output levels set by the global oil reduction deal, Falih added.
"I confirm our commitment to meet all these requests (to replace Iranian oil). But at the same time, we will do this remaining part of the OPEC+ deal, we will stick to it. We do not need to voluntarily exceed the limits set," he said.
The Organization of the Petroleum Exporting Countries (OPEC), Russia and other producers, an alliance known as OPEC+, agreed to cut output by 1.2 million barrels per day (bpd) from January for six months in an effort to boost oil prices.
The oil producers meet on June 25-26 to decide whether to extend the pact or adjust supply targets.
Earlier in April, Moscow signalled OPEC and its allies could raise oil output from June because of improving market conditions and falling stockpiles.
"We will look at (global oil) inventories - are they higher or lower than the normal level and we will adjust the production level accordingly. Based on what I see now ... I am eager to say there will be some kind of agreement," Falih RIA.
"It may remain the same, or could change up or down, I don't know."
One of the key players on Moscow's policy towards OPEC is Igor Sechin, head of Russian state oil company Rosneft .
Sechin, an ally of President Vladimir Putin, says Russia is losing market share to the United States, which is not participating in production cuts and has boosted output to record levels of some 12 million bpd.
Last week, Sechin signalled Russia would not help replace Iranian oil on the market after the expiration of waivers on US sanctions against Tehran's crude exports.
Iranian exports were not significant, Falih said.
"The only indicator I have is consumers' demand for Saudi oil ... These figures are moderate at the moment, the demand is healthy, there is nothing to worry about... There is no shortage on the (global oil) market," he said.
Saudi oil production until the end of May would be below the level set in the global deal: "significantly less" than 10 million bpd, with exports below 7 million bpd next month, Falih said. Under the OPEC+ deal, Riyadh can pump up to 10.3 million bpd.
"We are comfortable with the overall situation on the (global oil) market: it is healthy, it is well-supplied, nothing to worry about," he added.
A panel of energy ministers from major oil producers, including Saudi Arabia and Russia, known as the JMMC, meets on May 19 to discuss the oil market and make recommendations before the June meeting, OPEC sources said.