In basket trials a single therapeutic treatment is tested on several patient populations simultaneously, each of which forming a basket, where patients across all baskets on the trial share a common genetic aberration. These trials allow testing of treatments on small groups of patients, however, limited basket sample sizes can result in inadequate precision and power of estimates. It is well known that Bayesian information borrowing models such as the exchangeability-nonexchangeability (EXNEX) model can be implemented to tackle such a problem, drawing on information from one basket when making inference in another. An alternative approach to improve power of estimates, is to incorporate any historical or external information available. This paper considers models that amalgamate both forms of information borrowing, allowing borrowing between baskets in the ongoing trial whilst also drawing on response data from historical sources, with the aim to further improve treatment effect estimates. We propose several Bayesian information borrowing approaches that incorporate historical information into the model. These methods are data-driven, updating the degree of borrowing based on the level of homogeneity between information sources. A thorough simulation study is presented to draw comparisons between the proposed approaches, whilst also comparing to the standard EXNEX model in which no historical information is utilized. The models are also applied to a real-life trial example to demonstrate their performance in practice. We show that the incorporation of historic data under the novel approaches can lead to a substantial improvement in precision and power of treatment effect estimates when such data is homogeneous to the responses in the ongoing trial. Under some approaches, this came alongside an inflation in type I error rate in cases of heterogeneity. However, the use of a power prior in the EXNEX model is shown to increase power and precision, whilst maintaining similar error rates to the standard EXNEX model.