Saudi's July crude output is lower than its June production by around 200,000 bpd, the source said.
"We believe now after talking with customers and consumers that the market is getting tighter, as we see that the US commercial stocks are going down," the source said.
Saudi Arabia's production target under an output-cutting deal among the Organization of the Petroleum Exporting Countries and allies, a grouping known as OPEC+, is 10.3 million bpd.
The kingdom will continue to produce below that level in coming months, the source said, as it seeks to reduce oil inventories.
"Saudi Arabia has a strong commitment to the OPEC+ group decisions and is determined to see lower global oil stocks," the source said.