Is there a correlation between EQ (EI) and IQ? Is there a correlation, positive or negative, between emotional intelligence and IQ?
My understanding is that self-report measures of emotional intelligence tend to have very weak correlations with traditional measures of cognitive ability, and ability-based measures of emotional intelligence have low to moderate correlations with traditional measures of cognitive ability. The following provides relevant extracts from Conte (2005) that directly relates to these points. For a more complete picture you may wish to read the article.

Self-report measures of emotional intelligence


  The EQ-i is a 133-item self-report measure that takes approximately 30 minutes to complete (Bar-On, 2000) ...
  With respect to discriminant validity, the EQ-i correlated 0.12 with
  the Wechsler Adult Intelligence Scale (Bar-On, 2000), and the average
  correlation between the EQ-i and the Big Five personality measures was
  approximately 0.50 (Dawda &amp; Hart, 2000)


Ability based measures of emotional intelligence

The MEIS and MSCEIT are both ability based measures of emotional intelligence. 


  Data from several studies indicate that the MEIS correlates between 0.30 and 0.40 with traditional measures
  of cognitive ability (Roberts et al., 2001; Van Rooy &amp; Viswesvaran, 2004)


Relation between ability and personality based measures

It also appears that self-report and ability based measures of emotional intelligence are measuring relatively distinct variables. 


  Mayer et al. (2000) found that the MSCEIT and Bar-On scales correlated
  0.36, indicating that they share approximately 13 percent of their variance.
  In a more recent study, Brackett and Mayer (2003) found that
  the MSCEIT and Bar-On scales correlated 0.21, indicating that they
  share approximately 4 percent of their variance.


References


Bar-On, R. (2000). 
Emotional and Social Intelligence: Insights from the Emotional Quotient Inventory.
In Bar-On, R.; Parker, J. (eds.). 
The handbook of emotional intelligence: Theory, development, assessment, and application at home, school, and in the workplace (pp. 363–388). Jossey-Bass. 
ISBN 978-0-470-90743-6
Brackett, M. A., &amp; Mayer, J. D. (2003). 
Convergent, discriminant, and incremental validity of competing measures of emotional intelligence.
Personality and Social Psychology Bulletin, 29(9), 1147–1158. 
doi:10.1177/0146167203254596
Conte, J.M. (2005). 
A review and critique of emotional intelligence measures. 
Journal of Organizational Behavior, 26(4), 433–440. 
doi:10.1002/job.319. 
[PDF]
Dawda, D., &amp; Hart, S. D. (2000). 
Assessing emotional intelligence: reliability and validity of the Bar-On Emotional Quotient Inventory (EQ-i) in university students. 
Personality and Individual Differences, 28(4), 797–812. 
doi:10.1016/S0191-8869(99)00139-7
Mayer, J. D., Caruso, D., &amp; Salovey, P. (2000). 
Selecting a measure of emotional intelligence: the case for ability scales. 
In Bar-On, R.; Parker, J. (eds.). 
The handbook of emotional intelligence: Theory, development, assessment, and application at home, school, and in the workplace (pp. 320–342). Jossey-Bass. 
ISBN 978-0-470-90743-6
Roberts, R. D., Zeidner, M., &amp; Matthews, G. (2001). 
Does emotional intelligence meet traditional standards for an intelligence? Some new data and conclusions. 
Emotion, 1(3), 196–231. 
doi:10.1037/1528-3542.1.3.196
Van Rooy, D. L., &amp; Viswesvaran, C. (2004). 
Emotional intelligence: a meta-analytic investigation of predictive validity and nomological net. 
Journal of Vocational Behavior, 65(1), 71–95. 
doi:10.1016/S0001-8791(03)00076-9