The world is currently facing an outbreak of the monkeypox virus, and confirmed cases have been reported from 28 countries. Following a recent “emergency meeting”, the World Health Organization is considering whether the outbreak should be assessed as a “potential public health emergency of international concern” or PHEIC, as was done for the COVID-19 and Ebola outbreaks in the past. During this time, people from all over the world are using social media platforms, such as Twitter, for information seeking and sharing related to the outbreak, as well as for familiarizing themselves with the guidelines and protocols that are being recommended by various policy-making bodies to reduce the spread of the virus. This is resulting in the generation of tremendous amounts of Big Data related to such paradigms of social media behavior. Mining this Big Data and compiling it in the form of a dataset can serve as a data resource for a wide range of use-cases and applications such as analysis of public opinions, interests, views, perspectives, attitudes, and sentiment towards this outbreak. Therefore, this work presents MonkeyPox2022Tweets, an open-access dataset of Tweets related to the 2022 monkeypox outbreak that were posted on Twitter since the first detected case of this outbreak on May 7, 2022. The dataset is compliant with the privacy policy, developer agreement, and guidelines for content redistribution of Twitter, as well as with the FAIR principles (Findability, Accessibility, Interoperability, and Reusability) principles for scientific data management.