This isn't the easiest question to answer. You could look at transcription factor binding site position weight matrices like those from TRANSFAC and come up with a list of all factors that potentially hit that site, then perform some kind of enrichment analysis on that. But this involves some programming, and is based solely on sequence motifs, not experimental data.
The ENCODE consortium spent over $100M and generated hundreds of ChIP-seq experiments for different transcription factors and histone modifications across many cell types (if you don't know much about ENCODE, go read the main ENCODE paper, and Sean Eddy's very fair commentary). Regardless of what you might consider "biologically functional", the ENCODE project generated a ton of data, and much of this data is publicly available. But that still doesn't help answer our question, because genes are often bound by multiple TFs, and TFs can bind many regions. We need to perform an enrichment (read: hypergeometric) test to assess an over-representation of experimentally bound transcription factors around our gene targets of interest ("around" also implies that some spatial boundary must be specified). To date, I haven't found a good tool to do this easily.
Raymond Auerbach and Bin Chen in Atul Butte's lab recently developed a resource to address this very common need, called the ENCODE ChIP-Seq Significance Tool.
The paper: Auerbach et al. Relating Genes to Function: Identifying Enriched Transcription Factors using the ENCODE ChIP-Seq Significance Tool. Bioinformatics (2013): 10.1093/bioinformatics/btt316.
The software: ENCODE ChIP-Seq Significance Tool (http://encodeqt.stanford.edu/).
This tool takes a list of "interesting" (significant, dysregulated, etc.) genes as input, and identifies ENCODE transcription factors from this list. Head over to http://encodeqt.stanford.edu/, select the ID type you're using (Ensembl, Symbol, etc), and paste in your list of genes. You can also specify your background set (this has big implications for the significance testing using the hypergeometric distribution). Scroll down some more to tell the tool how far up and downstream you want to look from the transcription start/end site or whole gene, select an ENCODE cell line (or ALL), and hit submit.
You're then presented with a list of transcription factors that are most likely regulating your input genes (based on overrepresentation of ENCODE ChIP-seq binding sites). Your results can then be saved to CSV or PDF. You can also click on a number in the results table and get a list of genes that are regulated by a particular factor (the numbers do not appear as hyperlinks in my browser, but clicking the number still worked).
At the very bottom of the page, you can load example data that they used in the supplement of their paper, and run through the analysis presented therein. The lead author, Raymond Auerbach, even made a very informative screencast on how to use the tool:
Now, if I could only find a way to do something like this with mouse gene expression data.