After the first weeks of vaccination against the SARS-CoV-2, several cases of acute thrombosis were reported. These news reports began to be shared frequently across social media platforms. The aim of this study was to conduct an analysis of Twitter data related to the overall discussion. Data was retrieved from 14th March to 14th April using the keyword ‘blood clots’. A dataset with n=266,677 tweets was retrieved, and a systematic random sample of 5% of tweets (n=13,334) were entered into NodeXL for further analysis. Social network analysis was used to analyse the data by drawing upon the Clauset-Newman-Moore algorithm. Influential users were identified by drawing upon the betweenness centrality metric. Text analysis was applied to identify the key hashtags and websites used at this time. More than half of the network was comprised of retweets and the largest groups within the network were broadcast clusters where a number of key users were retweeted. The most popular narratives were around highlighting the low risk of obtaining a blood clot from a vaccine and highlighting higher blood clot risks in medicines commonly consumed. A wide-variety of actors drove the discussion on Twitter ranging from writers, physicians, the general public academics, celebrities, and journalists. Twitter was used to highlight the low potential of obtaining a blood clot from a vaccine and encouraged vaccinations among the public.