Speaking to US broadcaster CBS News, Qatari Emir Sheikh Tamim bin Hamad al-Thani said he wanted an end to the dispute, which erupted on June 5 and pits Saudi Arabia, Bahrain, the United Arab Emirates and Egypt against Qatar.
"Nothing is going to be above our dignity, our sovereignty. But we want it to end. I always say that," he told the "60 Minutes" programme in an interview aired on Sunday. "If they (are) going to walk one metre toward me, I'm willing to walk 10,000 miles towards them," he said.
The four countries have cut diplomatic, transport and trade ties with Qatar, the world's top seller of liquefied natural gas, accusing it of financing terrorism. Doha denies the charges.
The dispute has hit travel and food imports and ratcheted up tensions in a wealthy region which plays a leading role in global energy markets as well as events across the wider Middle East.
Qatar hosts Al Udeid air base, the largest US military facility in the Middle East.
Sheikh Tamim said Trump had told him "I will not accept my friends fighting amongst themselves," and that in talks on the sidelines of a United Nations' meeting in September he had made an offer to host talks in the United States.
"I told him straightaway, 'Mr President, we are very ready. I've been asking for dialogue from day one," Sheikh Tamim said, adding that the meeting was supposed to happen very soon and that he had not heard a response from the other countries.
President Trump, who has said he would be willing to mediate in the dispute, said in September he had a "very strong feeling" it would be solved "pretty quickly".
Earlier in September, Saudi Arabia suspended any dialogue with Qatar, accusing it of "distorting facts," just after a report of a phone call between the leaders of the two countries suggested a possible breakthrough in the dispute.
Saudi Arabia's Crown Prince Mohammed bin Salman spoke by phone with Qatar's emir on September 8 in the first publicly reported contact between the two leaders since the crisis began.
There has been no further contact reported since then.
In the CBS interview, Sheikh Tamim reiterated that Qatar would not close down the Doha-based Al Jazeera television network, as demanded by the four countries who accuse the broadcaster of bias and interfering in their affairs.
Al Jazeera says it is an independent news service giving a voice to everyone in the region.
Sheikh Tamim also said he feared for the region if any military actions were taken as part of the crisis.
"I'm fearful that if anything happens, if any military act happens, this region will be in chaos," he said.