Asthma CAN cause coughing.

More Info: Some asthmatics may experience coughing as a symptom of their asthma.  If coughing is the main symptom of your asthma, you may have cough-variant asthma (CVA). Those with CVA exhibit coughing as a main symptom rather than the typical wheezing and shortness of breath.

Cough-Variant Asthma Symptoms  WebMD

What Is Cough Variant Asthma?

Cough variant asthma is a dry cough that does not produce or expel any mucus. People who have cough variant asthma do not show any normal or common symptoms of asthma such as shortness of breath or wheezing. Another name for cough variant asthma is chronic cough. Normally the cough persists for six to eight weeks. Coughing can occur at night or day, with some people it worsens during physical activity, and it can disrupt sleep. Certain triggers such as allergens, cold air, dust or potent fragrances can make coughing worse. It is common in young children who have asthma, but anyone can get cough variant asthma. There are no known causes, but it can occur after an upper repository infection. Cough variant asthma is treated the same way as asthma including the use of an inhaler.              

How Is Cough Variant Asthma Treated?

The treatment of Cough Variant Asthma is essentially the same as the treatment of asthma. Relief is often found using bronchodilators, and in the case of bronchodilators relief comes rapidly, often within a week. If a patient does not find relief through bronchodilators, inhaled steroids are the next course of action. Complete relief is usually seen within eight weeks of starting steroid treatment. Steroids that are more potent are prescribed if the cough persists.